From 90df73f67a5071aaad16201d1075aad38d48ca56 Mon Sep 17 00:00:00 2001
From: Paula Gearon <gearon@gmail.com>
Date: Tue, 25 Apr 2023 01:29:02 -0400
Subject: [PATCH 01/12] Added example solutions

---
 doc/scripts/hashsets.clj | 90 ++++++++++++++++++++++++++++++++++++----
 doc/scripts/trees.clj    | 76 +++++++++++++++++++++++++++++----
 doc/scripts/vectors.clj  | 22 +++++++++-
 3 files changed, 173 insertions(+), 15 deletions(-)

diff --git a/doc/scripts/hashsets.clj b/doc/scripts/hashsets.clj
index 3d209a9..d153c74 100644
--- a/doc/scripts/hashsets.clj
+++ b/doc/scripts/hashsets.clj
@@ -124,7 +124,22 @@ words
 ;; #### Exercise
 ;; Rewrite `insert!` and `contains-object?` to use a linked list at each array location
 
-;; _Solution goes here_
+;; A Solution:
+(defn insert!
+  [arr s]
+  (let [pos (array-location (count arr) s)
+        existing-list (nth arr pos)]
+    (if (some #(= s %) existing-list) ;; search the list
+      arr  ;; already there, so just return the original array
+      (aset arr pos (cons s existing-list))))) ;; add to the list, and put this back in the array
+
+(defn contains-object?
+  [arr s]
+  ;; find the list at the location in the array
+  (let [found (nth arr (array-location (count arr) s))]
+    ;; search the list
+    (some #(= s %) found)))
+
 
 ;; Let's test this by repopulate the array of words, and adding "apple":
 (def words (object-array 10))
@@ -242,8 +257,10 @@ words50
 
 ;; #### Exercise
 ;; Rewrite `array-location` to use the `hash` function
-
-;; _Solution goes here_
+;; A Solution:
+(defn array-location
+  [size s]
+  (mod (hash s) size))
 
 ;; When using a REPL, the `insert` and `contains-object?` functions will pick up this new version of `array-location`
 ;; (defn insert!
@@ -293,7 +310,29 @@ words50
     (aset p 1 v)
     p))
 
-;; _Solution goes here_
+;; A Solution:
+(defn insert-map!
+  [arr k v]
+  ;; get the location in the array
+  (let [pos (array-location (count arr) k)
+        ;; get the existing data at that location
+        existing (nth arr pos)]
+    ;; check the first of each pair, to see if it matches the key
+    (if-let [[fk fv] (first (filter #(= k (first %)) existing))]
+      ;; see if the value is changing
+      (if (= fv v)
+        arr  ;; no change
+        ;; otherwise, remove the old pair, and add the new one
+        (aset arr pos (cons (pair k v) (remove #(= k (first %)) existing))))
+      ;; the key is not there, so just add the pair
+      (aset arr pos (cons (pair k v) existing)))))
+
+(defn get-map
+  [arr k]
+  ;; find the existing list of pairs
+  (let [found (nth arr (array-location (count arr) k))]
+    ;; search the list for the matching key, then return the value
+    (second (first (filter #(= k (first %)) found)))))
 
 ;; Let's test this out with an array of size 50:
 (def map-array (object-array 50))
@@ -321,7 +360,10 @@ map-array
 ;; Write a `to-seq` function that will convert the hashmap array to a seq of pairs.
 ;; Convert these pairs to 2 element vectors.
 
-;; _Solution goes here_
+;; A Solution:
+(defn to-seq
+  [arr]
+  (mapcat #(map vec %) arr))
 
 ;; Apply this to the map-array:
 (to-seq map-array)
@@ -330,7 +372,16 @@ map-array
 ;; Write a `map-keys` function and a `map-vals` function to duplicate the
 ;; `keys` and `vals` functions for maps:
 
-;; _Solution goes here_
+;; A Solution:
+(defn map-keys
+  [arr]
+  (map first (to-seq arr)))
+
+(defn map-vals
+  [arr]
+  (map second (to-seq arr)))
+
+;; Not all of the exercises have to be hard!
 
 ;; ## Immutability
 ;; In order to follow how to make these structures immutable, let's turn to Tree structures.
@@ -345,7 +396,32 @@ map-array
 ;; Write the functions `insert` and `get-map` to duplicate the previous functionality, but
 ;; now the hashmap is in a vector, and the operations must be immutable.
 
-;; _Solution goes here_
+;; A solution. This includes all operations:
+
+(defn location
+  [size s]
+  (mod (hash s) size))
+
+(defn insert
+  ;; accept a key/value as a pair
+  ([vctr [k v]] (insert vctr k v))
+  ;; accept the key and value as separate arguments
+  ([vctr k v]
+   (let [pos (location (count vctr) k)
+         existing (nth vctr pos)]
+     (if-let [[fk fv] (first (filter #(= k (first %)) existing))]
+       (if (= fv v)
+         vctr
+         (assoc vctr pos (cons [k v] (remove #(= k (first %)) existing))))
+       (assoc vctr pos (cons [k v] existing))))))
+
+(defn get-map
+  [vctr k]
+  (let [found (nth vctr (location (count vctr) k))]
+    (second (first (filter #(= k (first %)) found)))))
+
+;; Neither the `location` not the `get-map` function needed to change.
+;; only the `insert` function changed, to use `assoc` rather than `aset`
 
 ;; Initialize a vector at the full width, full of nils:
 (def vctr (vec (repeat 50 nil)))
diff --git a/doc/scripts/trees.clj b/doc/scripts/trees.clj
index ef65af9..3beead5 100644
--- a/doc/scripts/trees.clj
+++ b/doc/scripts/trees.clj
@@ -93,7 +93,30 @@
 
 [[5, [1, nil, [3, nil, nil]], [9, nil, nil]]]
 
-;; _Solution goes here_
+
+;; A Solution:
+(defn put!
+  [tree value]
+  (if-not (root tree)
+    ;; if empty, create a new tree with just one node
+    (set-root! tree (node value))
+    ;; iterate down the branches
+    (loop [current-node (root tree)]
+      ;; read the number at the current node
+      (let [current-value (data current-node)]
+        (if (= current-value value)
+          ;; when equal to the number being inserted, then do nothing and return.
+          tree
+          ;; determine the side. Smaller goes to the left, larger to the right.
+          (let [side (if (< value current-value) :left :right)]
+            (if-let [child-node (child current-node side)]
+              ;; if a child node exists on that side, step down and repeat
+              (recur child-node)
+              ;; no child on the selected side, so create a leaf node and insert it
+              (do
+                (set-child! current-node side (node value))
+                ;; return the root of the tree
+                tree))))))))
 
 ;; Try it out...
 
@@ -114,7 +137,17 @@
 ;; Calling `(t-seq t)` should return `(1 3 5 9)`
 
 
-;; _Solution goes here_
+;; A Solution:
+(defn as-seq
+  [node]
+  (and node
+       (concat (as-seq (left node))
+               [(data node)]
+               (as-seq (right node))))) 
+
+(defn t-seq
+  [tree]
+  (as-seq (root tree))) 
 
 (t-seq t)
 
@@ -177,13 +210,24 @@
 ;; #### Exercise:
 ;; Write a function `utree` that creates a tree from a seq parameter.
 
-;; _Solution goes here_
+;; A solution:
+(defn utree [s] (reduce put nil s))
 
 ;; #### Exercise:
 ;; Write a function called `tcontains?` which checks if a value has been stored in the tree. This is the same operation
 ;; as `clojure.core/contains?`
 
-;; _Solution goes here_
+;; A solution:
+(defn tcontains?
+  [node value]
+  (and node   ;; test if the tree rooted at `node` is not empty
+       ;; look at the data for the current node
+       (let [v (data node)]
+         (or
+           ;; if the data node data is the same as what is being looked for, then return true
+          (= value v)
+          ;; otherwise, recurse into the left child branch if the value is smaller, or the right if larger
+          (recur (if (< value v) (left n) (right node)))))))
 
 (def t (utree [5 1 9]))
 (tcontains? t 1)
@@ -315,7 +359,12 @@
 ;; #### Exercise:
 ;; Write a `max-depth` function to return the maximum depth of the rb-tree. Hint: use recursion.
 
-;; _Solution goes here_
+;; A Solution:
+(defn max-depth
+  [n]
+  (if n
+    (inc (max (max-depth (left n)) (max-depth (right n))))
+    0))
 
 ;; The maximum depth can indicate that a tree is relatively balanced. A perfectly balanced tree
 ;; ought to be about the log2 of the number of items. Red/Black trees do not balance perfectly, but we should
@@ -381,7 +430,18 @@
 ;; Write `tget`: a function that adds to the `tcontains?` function to search for a given key,
 ;; returning the associated value.
 
-;; _Solution goes here_
+;; A solution:
+(defn tget
+  [node key]
+  ;; return nil for an empty tree
+  (and node
+       ;; compare the value at this node to the key being searched for
+       (let [c (compare key (nkey node))]
+         (if (zero? c)
+           ;; when the node contains the required key, return the associated value
+           (value node)
+           ;; key not at this node, so look in the children, going left for smaller, right for larger.
+           (recur (if (< c 0) (left node) (right node)))))))
 
 ;; The following map of labels to numbers can be used to demonstrate this function:
 
@@ -403,7 +463,9 @@
 ;; Implement the `tkeys` and `tvals` functions that return the keys and values of this tree set as seqs.
 ;; Hint: you can use the `as-seq` from above.
 
-;; _Solution goes here_
+;; A Solution:
+(defn tkeys [tree] (map first (as-seq tree)))
+(defn tvals [tree] (map second (as-seq tree)))
 
 (tkeys numbers)
 (tvals numbers)
diff --git a/doc/scripts/vectors.clj b/doc/scripts/vectors.clj
index 5b0691c..5c7b29f 100644
--- a/doc/scripts/vectors.clj
+++ b/doc/scripts/vectors.clj
@@ -152,7 +152,27 @@ v
 ;; Write a function called vth that can manually retrieve the nth element from a vector that contains
 ;; between 1057 and 32800 elements. i.e. There are 3 levels.
 
-;; _Solution goes here_
+;; A Solution:
+(defn vth
+  [v n]
+  ;; check that the offset is not out of range
+  (when (< n (count v))
+    ;; get the offset in the root
+    (let [first-level (int (/ n 1024))
+          ;; record the remaining offset
+          n2 (mod n 1024)
+          ;; the offset in the second level of the tree
+          second-level (int (/ n2 32))
+          ;; the final offset
+          third-level (mod n2 32)]
+      ;; check if the offset is beyond the range of the first level
+      (if (or (> first-level 32)
+              ;; or the second level offset is a nil node
+              (nil? (-> v .root .array (nth first-level) .array (nth second-level))))
+        ;; this indicates that the offset is beyond the current data and inside the tail
+        (-> v .tail (nth third-level))
+        ;; otherwise, traverse down the tree to the offset
+        (-> v .root .array (nth first-level) .array (nth second-level) .array (nth third-level))))))
 
 (vth vec1057 500)
 ;; 500

From 5b677a3ada00418ba1e9fc29c5a34fcbbf2fd4f6 Mon Sep 17 00:00:00 2001
From: Paula Gearon <gearon@gmail.com>
Date: Tue, 25 Apr 2023 01:29:02 -0400
Subject: [PATCH 02/12] Added example solutions

---
 doc/scripts/hashsets.clj | 90 ++++++++++++++++++++++++++++++++++++----
 doc/scripts/trees.clj    | 76 +++++++++++++++++++++++++++++----
 doc/scripts/vectors.clj  | 22 +++++++++-
 3 files changed, 173 insertions(+), 15 deletions(-)

diff --git a/doc/scripts/hashsets.clj b/doc/scripts/hashsets.clj
index 3d209a9..d153c74 100644
--- a/doc/scripts/hashsets.clj
+++ b/doc/scripts/hashsets.clj
@@ -124,7 +124,22 @@ words
 ;; #### Exercise
 ;; Rewrite `insert!` and `contains-object?` to use a linked list at each array location
 
-;; _Solution goes here_
+;; A Solution:
+(defn insert!
+  [arr s]
+  (let [pos (array-location (count arr) s)
+        existing-list (nth arr pos)]
+    (if (some #(= s %) existing-list) ;; search the list
+      arr  ;; already there, so just return the original array
+      (aset arr pos (cons s existing-list))))) ;; add to the list, and put this back in the array
+
+(defn contains-object?
+  [arr s]
+  ;; find the list at the location in the array
+  (let [found (nth arr (array-location (count arr) s))]
+    ;; search the list
+    (some #(= s %) found)))
+
 
 ;; Let's test this by repopulate the array of words, and adding "apple":
 (def words (object-array 10))
@@ -242,8 +257,10 @@ words50
 
 ;; #### Exercise
 ;; Rewrite `array-location` to use the `hash` function
-
-;; _Solution goes here_
+;; A Solution:
+(defn array-location
+  [size s]
+  (mod (hash s) size))
 
 ;; When using a REPL, the `insert` and `contains-object?` functions will pick up this new version of `array-location`
 ;; (defn insert!
@@ -293,7 +310,29 @@ words50
     (aset p 1 v)
     p))
 
-;; _Solution goes here_
+;; A Solution:
+(defn insert-map!
+  [arr k v]
+  ;; get the location in the array
+  (let [pos (array-location (count arr) k)
+        ;; get the existing data at that location
+        existing (nth arr pos)]
+    ;; check the first of each pair, to see if it matches the key
+    (if-let [[fk fv] (first (filter #(= k (first %)) existing))]
+      ;; see if the value is changing
+      (if (= fv v)
+        arr  ;; no change
+        ;; otherwise, remove the old pair, and add the new one
+        (aset arr pos (cons (pair k v) (remove #(= k (first %)) existing))))
+      ;; the key is not there, so just add the pair
+      (aset arr pos (cons (pair k v) existing)))))
+
+(defn get-map
+  [arr k]
+  ;; find the existing list of pairs
+  (let [found (nth arr (array-location (count arr) k))]
+    ;; search the list for the matching key, then return the value
+    (second (first (filter #(= k (first %)) found)))))
 
 ;; Let's test this out with an array of size 50:
 (def map-array (object-array 50))
@@ -321,7 +360,10 @@ map-array
 ;; Write a `to-seq` function that will convert the hashmap array to a seq of pairs.
 ;; Convert these pairs to 2 element vectors.
 
-;; _Solution goes here_
+;; A Solution:
+(defn to-seq
+  [arr]
+  (mapcat #(map vec %) arr))
 
 ;; Apply this to the map-array:
 (to-seq map-array)
@@ -330,7 +372,16 @@ map-array
 ;; Write a `map-keys` function and a `map-vals` function to duplicate the
 ;; `keys` and `vals` functions for maps:
 
-;; _Solution goes here_
+;; A Solution:
+(defn map-keys
+  [arr]
+  (map first (to-seq arr)))
+
+(defn map-vals
+  [arr]
+  (map second (to-seq arr)))
+
+;; Not all of the exercises have to be hard!
 
 ;; ## Immutability
 ;; In order to follow how to make these structures immutable, let's turn to Tree structures.
@@ -345,7 +396,32 @@ map-array
 ;; Write the functions `insert` and `get-map` to duplicate the previous functionality, but
 ;; now the hashmap is in a vector, and the operations must be immutable.
 
-;; _Solution goes here_
+;; A solution. This includes all operations:
+
+(defn location
+  [size s]
+  (mod (hash s) size))
+
+(defn insert
+  ;; accept a key/value as a pair
+  ([vctr [k v]] (insert vctr k v))
+  ;; accept the key and value as separate arguments
+  ([vctr k v]
+   (let [pos (location (count vctr) k)
+         existing (nth vctr pos)]
+     (if-let [[fk fv] (first (filter #(= k (first %)) existing))]
+       (if (= fv v)
+         vctr
+         (assoc vctr pos (cons [k v] (remove #(= k (first %)) existing))))
+       (assoc vctr pos (cons [k v] existing))))))
+
+(defn get-map
+  [vctr k]
+  (let [found (nth vctr (location (count vctr) k))]
+    (second (first (filter #(= k (first %)) found)))))
+
+;; Neither the `location` not the `get-map` function needed to change.
+;; only the `insert` function changed, to use `assoc` rather than `aset`
 
 ;; Initialize a vector at the full width, full of nils:
 (def vctr (vec (repeat 50 nil)))
diff --git a/doc/scripts/trees.clj b/doc/scripts/trees.clj
index 6f969bb..7a2537e 100644
--- a/doc/scripts/trees.clj
+++ b/doc/scripts/trees.clj
@@ -93,7 +93,30 @@
 
 [[5, [1, nil, [3, nil, nil]], [9, nil, nil]]]
 
-;; _Solution goes here_
+
+;; A Solution:
+(defn put!
+  [tree value]
+  (if-not (root tree)
+    ;; if empty, create a new tree with just one node
+    (set-root! tree (node value))
+    ;; iterate down the branches
+    (loop [current-node (root tree)]
+      ;; read the number at the current node
+      (let [current-value (data current-node)]
+        (if (= current-value value)
+          ;; when equal to the number being inserted, then do nothing and return.
+          tree
+          ;; determine the side. Smaller goes to the left, larger to the right.
+          (let [side (if (< value current-value) :left :right)]
+            (if-let [child-node (child current-node side)]
+              ;; if a child node exists on that side, step down and repeat
+              (recur child-node)
+              ;; no child on the selected side, so create a leaf node and insert it
+              (do
+                (set-child! current-node side (node value))
+                ;; return the root of the tree
+                tree))))))))
 
 ;; Try it out...
 
@@ -114,7 +137,17 @@
 ;; Calling `(t-seq t)` should return `(1 3 5 9)`
 
 
-;; _Solution goes here_
+;; A Solution:
+(defn as-seq
+  [node]
+  (and node
+       (concat (as-seq (left node))
+               [(data node)]
+               (as-seq (right node))))) 
+
+(defn t-seq
+  [tree]
+  (as-seq (root tree))) 
 
 (t-seq t)
 
@@ -177,13 +210,24 @@
 ;; #### Exercise:
 ;; Write a function `utree` that creates a tree from a seq parameter.
 
-;; _Solution goes here_
+;; A solution:
+(defn utree [s] (reduce put nil s))
 
 ;; #### Exercise:
 ;; Write a function called `tcontains?` which checks if a value has been stored in the tree. This is the same operation
 ;; as `clojure.core/contains?`
 
-;; _Solution goes here_
+;; A solution:
+(defn tcontains?
+  [node value]
+  (and node   ;; test if the tree rooted at `node` is not empty
+       ;; look at the data for the current node
+       (let [v (data node)]
+         (or
+           ;; if the data node data is the same as what is being looked for, then return true
+          (= value v)
+          ;; otherwise, recurse into the left child branch if the value is smaller, or the right if larger
+          (recur (if (< value v) (left n) (right node)))))))
 
 (def t (utree [5 1 9]))
 (tcontains? t 1)
@@ -315,7 +359,12 @@
 ;; #### Exercise:
 ;; Write a `max-depth` function to return the maximum depth of the rb-tree. Hint: use recursion.
 
-;; _Solution goes here_
+;; A Solution:
+(defn max-depth
+  [n]
+  (if n
+    (inc (max (max-depth (left n)) (max-depth (right n))))
+    0))
 
 ;; The maximum depth can indicate that a tree is relatively balanced. A perfectly balanced tree
 ;; ought to be about the log2 of the number of items. Red/Black trees do not balance perfectly, but we should
@@ -381,7 +430,18 @@
 ;; Write `tget`: a function that adds to the `tcontains?` function to search for a given key,
 ;; returning the associated value.
 
-;; _Solution goes here_
+;; A solution:
+(defn tget
+  [node key]
+  ;; return nil for an empty tree
+  (and node
+       ;; compare the value at this node to the key being searched for
+       (let [c (compare key (nkey node))]
+         (if (zero? c)
+           ;; when the node contains the required key, return the associated value
+           (value node)
+           ;; key not at this node, so look in the children, going left for smaller, right for larger.
+           (recur (if (< c 0) (left node) (right node)))))))
 
 ;; The following map of labels to numbers can be used to demonstrate this function:
 
@@ -403,7 +463,9 @@
 ;; Implement the `tkeys` and `tvals` functions that return the keys and values of this tree set as seqs.
 ;; Hint: you can use the `as-seq` from above.
 
-;; _Solution goes here_
+;; A Solution:
+(defn tkeys [tree] (map first (as-seq tree)))
+(defn tvals [tree] (map second (as-seq tree)))
 
 (tkeys numbers)
 (tvals numbers)
diff --git a/doc/scripts/vectors.clj b/doc/scripts/vectors.clj
index 5b0691c..5c7b29f 100644
--- a/doc/scripts/vectors.clj
+++ b/doc/scripts/vectors.clj
@@ -152,7 +152,27 @@ v
 ;; Write a function called vth that can manually retrieve the nth element from a vector that contains
 ;; between 1057 and 32800 elements. i.e. There are 3 levels.
 
-;; _Solution goes here_
+;; A Solution:
+(defn vth
+  [v n]
+  ;; check that the offset is not out of range
+  (when (< n (count v))
+    ;; get the offset in the root
+    (let [first-level (int (/ n 1024))
+          ;; record the remaining offset
+          n2 (mod n 1024)
+          ;; the offset in the second level of the tree
+          second-level (int (/ n2 32))
+          ;; the final offset
+          third-level (mod n2 32)]
+      ;; check if the offset is beyond the range of the first level
+      (if (or (> first-level 32)
+              ;; or the second level offset is a nil node
+              (nil? (-> v .root .array (nth first-level) .array (nth second-level))))
+        ;; this indicates that the offset is beyond the current data and inside the tail
+        (-> v .tail (nth third-level))
+        ;; otherwise, traverse down the tree to the offset
+        (-> v .root .array (nth first-level) .array (nth second-level) .array (nth third-level))))))
 
 (vth vec1057 500)
 ;; 500

From 7df5ceb6b363756e09d48bf6f79426d3f91ac017 Mon Sep 17 00:00:00 2001
From: Paula Gearon <gearon@gmail.com>
Date: Wed, 26 Apr 2023 11:01:09 -0400
Subject: [PATCH 03/12] added missing param

---
 doc/scripts/trees.clj | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/doc/scripts/trees.clj b/doc/scripts/trees.clj
index 7a2537e..a9444fe 100644
--- a/doc/scripts/trees.clj
+++ b/doc/scripts/trees.clj
@@ -227,7 +227,7 @@
            ;; if the data node data is the same as what is being looked for, then return true
           (= value v)
           ;; otherwise, recurse into the left child branch if the value is smaller, or the right if larger
-          (recur (if (< value v) (left n) (right node)))))))
+          (recur (if (< value v) (left n) (right node))) value))))
 
 (def t (utree [5 1 9]))
 (tcontains? t 1)

From 70ca7cafb6bd72e8b00854d4ac92f21cdea2a015 Mon Sep 17 00:00:00 2001
From: Paula Gearon <gearon@gmail.com>
Date: Wed, 26 Apr 2023 11:22:00 -0400
Subject: [PATCH 04/12] added missing parameter

---
 doc/scripts/trees.clj | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/doc/scripts/trees.clj b/doc/scripts/trees.clj
index a9444fe..a537f69 100644
--- a/doc/scripts/trees.clj
+++ b/doc/scripts/trees.clj
@@ -441,7 +441,7 @@
            ;; when the node contains the required key, return the associated value
            (value node)
            ;; key not at this node, so look in the children, going left for smaller, right for larger.
-           (recur (if (< c 0) (left node) (right node)))))))
+           (recur (if (< c 0) (left node) (right node)) key)))))
 
 ;; The following map of labels to numbers can be used to demonstrate this function:
 

From 99c51527702607ee13e57a8e467d4119ded62058 Mon Sep 17 00:00:00 2001
From: Paula Gearon <gearon@gmail.com>
Date: Wed, 26 Apr 2023 11:24:31 -0400
Subject: [PATCH 05/12] Updated comment on Java HashSets

---
 doc/scripts/trees.clj | 7 ++++---
 1 file changed, 4 insertions(+), 3 deletions(-)

diff --git a/doc/scripts/trees.clj b/doc/scripts/trees.clj
index a537f69..32b73a0 100644
--- a/doc/scripts/trees.clj
+++ b/doc/scripts/trees.clj
@@ -470,8 +470,9 @@
 (tkeys numbers)
 (tvals numbers)
 
-;; Remember that in the Java memory model, the string does not appear twice in the array. Instead,
-;; _reference_ to the string is stored twice. This map of objects to themselves is the approach
-;; that Java uses for implementing Hashsets.
+;; Another approach to sets is to use maps and associate keys to themselves. This makes both the
+;; key and the value the same thing. Remember that in the Java memory model, the string does not
+;; appear twice in the array. Instead, a _reference_ to the string is stored twice, which is less
+;; expensive.. This map of objects to themselves is the approach that Java uses for implementing Hashsets.
 
 ;; Now that we know about immutable trees, we can build Vectors

From 833f7e9a291adc715ed08df9dfb35b19c38345cf Mon Sep 17 00:00:00 2001
From: Paula Gearon <gearon@gmail.com>
Date: Wed, 26 Apr 2023 11:27:51 -0400
Subject: [PATCH 06/12] Removed extra period

---
 doc/scripts/trees.clj | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/doc/scripts/trees.clj b/doc/scripts/trees.clj
index 32b73a0..1a27d12 100644
--- a/doc/scripts/trees.clj
+++ b/doc/scripts/trees.clj
@@ -473,6 +473,6 @@
 ;; Another approach to sets is to use maps and associate keys to themselves. This makes both the
 ;; key and the value the same thing. Remember that in the Java memory model, the string does not
 ;; appear twice in the array. Instead, a _reference_ to the string is stored twice, which is less
-;; expensive.. This map of objects to themselves is the approach that Java uses for implementing Hashsets.
+;; expensive. This map of objects to themselves is the approach that Java uses for implementing Hashsets.
 
 ;; Now that we know about immutable trees, we can build Vectors

From f99d5d525aaf93692616b789ea0ef9e33a4c7cc3 Mon Sep 17 00:00:00 2001
From: Paula Gearon <gearon@gmail.com>
Date: Tue, 25 Apr 2023 01:29:02 -0400
Subject: [PATCH 07/12] Added example solutions

---
 doc/scripts/hashsets.clj | 90 ++++++++++++++++++++++++++++++++++++----
 doc/scripts/trees.clj    | 76 +++++++++++++++++++++++++++++----
 doc/scripts/vectors.clj  | 22 +++++++++-
 3 files changed, 173 insertions(+), 15 deletions(-)

diff --git a/doc/scripts/hashsets.clj b/doc/scripts/hashsets.clj
index dfb9825..717138e 100644
--- a/doc/scripts/hashsets.clj
+++ b/doc/scripts/hashsets.clj
@@ -124,7 +124,22 @@ words
 ;; #### Exercise
 ;; Rewrite `insert!` and `contains-object?` to use a linked list at each array location
 
-;; _Solution goes here_
+;; A Solution:
+(defn insert!
+  [arr s]
+  (let [pos (array-location (count arr) s)
+        existing-list (nth arr pos)]
+    (if (some #(= s %) existing-list) ;; search the list
+      arr  ;; already there, so just return the original array
+      (aset arr pos (cons s existing-list))))) ;; add to the list, and put this back in the array
+
+(defn contains-object?
+  [arr s]
+  ;; find the list at the location in the array
+  (let [found (nth arr (array-location (count arr) s))]
+    ;; search the list
+    (some #(= s %) found)))
+
 
 ;; Let's test this by repopulate the array of words, and adding "apple":
 (def words (object-array 10))
@@ -242,8 +257,10 @@ words50
 
 ;; #### Exercise
 ;; Rewrite `array-location` to use the `hash` function
-
-;; _Solution goes here_
+;; A Solution:
+(defn array-location
+  [size s]
+  (mod (hash s) size))
 
 ;; When using a REPL, the `insert` and `contains-object?` functions will pick up this new version of `array-location`
 ;; (defn insert!
@@ -293,7 +310,29 @@ words50
     (aset p 1 v)
     p))
 
-;; _Solution goes here_
+;; A Solution:
+(defn insert-map!
+  [arr k v]
+  ;; get the location in the array
+  (let [pos (array-location (count arr) k)
+        ;; get the existing data at that location
+        existing (nth arr pos)]
+    ;; check the first of each pair, to see if it matches the key
+    (if-let [[fk fv] (first (filter #(= k (first %)) existing))]
+      ;; see if the value is changing
+      (if (= fv v)
+        arr  ;; no change
+        ;; otherwise, remove the old pair, and add the new one
+        (aset arr pos (cons (pair k v) (remove #(= k (first %)) existing))))
+      ;; the key is not there, so just add the pair
+      (aset arr pos (cons (pair k v) existing)))))
+
+(defn get-map
+  [arr k]
+  ;; find the existing list of pairs
+  (let [found (nth arr (array-location (count arr) k))]
+    ;; search the list for the matching key, then return the value
+    (second (first (filter #(= k (first %)) found)))))
 
 ;; Let's test this out with an array of size 50:
 (def map-array (object-array 50))
@@ -321,7 +360,10 @@ map-array
 ;; Write a `to-seq` function that will convert the hashmap array to a seq of pairs.
 ;; Convert these pairs to 2 element vectors.
 
-;; _Solution goes here_
+;; A Solution:
+(defn to-seq
+  [arr]
+  (mapcat #(map vec %) arr))
 
 ;; Apply this to the map-array:
 (to-seq map-array)
@@ -330,7 +372,16 @@ map-array
 ;; Write a `map-keys` function and a `map-vals` function to duplicate the
 ;; `keys` and `vals` functions for maps:
 
-;; _Solution goes here_
+;; A Solution:
+(defn map-keys
+  [arr]
+  (map first (to-seq arr)))
+
+(defn map-vals
+  [arr]
+  (map second (to-seq arr)))
+
+;; Not all of the exercises have to be hard!
 
 ;; ## Immutability
 ;; In order to follow how to make these structures immutable, let's turn to Tree structures.
@@ -345,7 +396,32 @@ map-array
 ;; Write the functions `insert` and `get-map` to duplicate the previous functionality, but
 ;; now the hashmap is in a vector, and the operations must be immutable.
 
-;; _Solution goes here_
+;; A solution. This includes all operations:
+
+(defn location
+  [size s]
+  (mod (hash s) size))
+
+(defn insert
+  ;; accept a key/value as a pair
+  ([vctr [k v]] (insert vctr k v))
+  ;; accept the key and value as separate arguments
+  ([vctr k v]
+   (let [pos (location (count vctr) k)
+         existing (nth vctr pos)]
+     (if-let [[fk fv] (first (filter #(= k (first %)) existing))]
+       (if (= fv v)
+         vctr
+         (assoc vctr pos (cons [k v] (remove #(= k (first %)) existing))))
+       (assoc vctr pos (cons [k v] existing))))))
+
+(defn get-map
+  [vctr k]
+  (let [found (nth vctr (location (count vctr) k))]
+    (second (first (filter #(= k (first %)) found)))))
+
+;; Neither the `location` not the `get-map` function needed to change.
+;; only the `insert` function changed, to use `assoc` rather than `aset`
 
 ;; Initialize a vector at the full width, full of nils:
 (def vctr (vec (repeat 50 nil)))
diff --git a/doc/scripts/trees.clj b/doc/scripts/trees.clj
index d492a5a..986a761 100644
--- a/doc/scripts/trees.clj
+++ b/doc/scripts/trees.clj
@@ -93,7 +93,30 @@
 
 [[5, [1, nil, [3, nil, nil]], [9, nil, nil]]]
 
-;; _Solution goes here_
+
+;; A Solution:
+(defn put!
+  [tree value]
+  (if-not (root tree)
+    ;; if empty, create a new tree with just one node
+    (set-root! tree (node value))
+    ;; iterate down the branches
+    (loop [current-node (root tree)]
+      ;; read the number at the current node
+      (let [current-value (data current-node)]
+        (if (= current-value value)
+          ;; when equal to the number being inserted, then do nothing and return.
+          tree
+          ;; determine the side. Smaller goes to the left, larger to the right.
+          (let [side (if (< value current-value) :left :right)]
+            (if-let [child-node (child current-node side)]
+              ;; if a child node exists on that side, step down and repeat
+              (recur child-node)
+              ;; no child on the selected side, so create a leaf node and insert it
+              (do
+                (set-child! current-node side (node value))
+                ;; return the root of the tree
+                tree))))))))
 
 ;; Try it out...
 
@@ -114,7 +137,17 @@
 ;; Calling `(t-seq t)` should return `(1 3 5 9)`
 
 
-;; _Solution goes here_
+;; A Solution:
+(defn as-seq
+  [node]
+  (and node
+       (concat (as-seq (left node))
+               [(data node)]
+               (as-seq (right node))))) 
+
+(defn t-seq
+  [tree]
+  (as-seq (root tree))) 
 
 (t-seq t)
 
@@ -177,13 +210,24 @@
 ;; #### Exercise:
 ;; Write a function `utree` that creates a tree from a seq parameter.
 
-;; _Solution goes here_
+;; A solution:
+(defn utree [s] (reduce put nil s))
 
 ;; #### Exercise:
 ;; Write a function called `tcontains?` which checks if a value has been stored in the tree. This is the same operation
 ;; as `clojure.core/contains?`
 
-;; _Solution goes here_
+;; A solution:
+(defn tcontains?
+  [node value]
+  (and node   ;; test if the tree rooted at `node` is not empty
+       ;; look at the data for the current node
+       (let [v (data node)]
+         (or
+           ;; if the data node data is the same as what is being looked for, then return true
+          (= value v)
+          ;; otherwise, recurse into the left child branch if the value is smaller, or the right if larger
+          (recur (if (< value v) (left n) (right node)))))))
 
 (def t (utree [5 1 9]))
 (tcontains? t 1)
@@ -315,7 +359,12 @@
 ;; #### Exercise:
 ;; Write a `max-depth` function to return the maximum depth of the rb-tree. Hint: use recursion.
 
-;; _Solution goes here_
+;; A Solution:
+(defn max-depth
+  [n]
+  (if n
+    (inc (max (max-depth (left n)) (max-depth (right n))))
+    0))
 
 ;; The maximum depth can indicate that a tree is relatively balanced. A perfectly balanced tree
 ;; ought to be about the log2 of the number of items. Red/Black trees do not balance perfectly, but we should
@@ -381,7 +430,18 @@
 ;; Write `tget`: a function that adds to the `tcontains?` function to search for a given key,
 ;; returning the associated value.
 
-;; _Solution goes here_
+;; A solution:
+(defn tget
+  [node key]
+  ;; return nil for an empty tree
+  (and node
+       ;; compare the value at this node to the key being searched for
+       (let [c (compare key (nkey node))]
+         (if (zero? c)
+           ;; when the node contains the required key, return the associated value
+           (value node)
+           ;; key not at this node, so look in the children, going left for smaller, right for larger.
+           (recur (if (< c 0) (left node) (right node)))))))
 
 ;; The following map of labels to numbers can be used to demonstrate this function:
 
@@ -403,7 +463,9 @@
 ;; Implement the `tkeys` and `tvals` functions that return the keys and values of this tree set as seqs.
 ;; Hint: you can use the `as-seq` from above.
 
-;; _Solution goes here_
+;; A Solution:
+(defn tkeys [tree] (map first (as-seq tree)))
+(defn tvals [tree] (map second (as-seq tree)))
 
 (tkeys numbers)
 (tvals numbers)
diff --git a/doc/scripts/vectors.clj b/doc/scripts/vectors.clj
index 5b0691c..5c7b29f 100644
--- a/doc/scripts/vectors.clj
+++ b/doc/scripts/vectors.clj
@@ -152,7 +152,27 @@ v
 ;; Write a function called vth that can manually retrieve the nth element from a vector that contains
 ;; between 1057 and 32800 elements. i.e. There are 3 levels.
 
-;; _Solution goes here_
+;; A Solution:
+(defn vth
+  [v n]
+  ;; check that the offset is not out of range
+  (when (< n (count v))
+    ;; get the offset in the root
+    (let [first-level (int (/ n 1024))
+          ;; record the remaining offset
+          n2 (mod n 1024)
+          ;; the offset in the second level of the tree
+          second-level (int (/ n2 32))
+          ;; the final offset
+          third-level (mod n2 32)]
+      ;; check if the offset is beyond the range of the first level
+      (if (or (> first-level 32)
+              ;; or the second level offset is a nil node
+              (nil? (-> v .root .array (nth first-level) .array (nth second-level))))
+        ;; this indicates that the offset is beyond the current data and inside the tail
+        (-> v .tail (nth third-level))
+        ;; otherwise, traverse down the tree to the offset
+        (-> v .root .array (nth first-level) .array (nth second-level) .array (nth third-level))))))
 
 (vth vec1057 500)
 ;; 500

From 39870c018824f6f61debd3ec10d5f9495e921616 Mon Sep 17 00:00:00 2001
From: Paula Gearon <gearon@gmail.com>
Date: Wed, 26 Apr 2023 11:01:09 -0400
Subject: [PATCH 08/12] added missing param

---
 doc/scripts/trees.clj | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/doc/scripts/trees.clj b/doc/scripts/trees.clj
index 986a761..db56e51 100644
--- a/doc/scripts/trees.clj
+++ b/doc/scripts/trees.clj
@@ -227,7 +227,7 @@
            ;; if the data node data is the same as what is being looked for, then return true
           (= value v)
           ;; otherwise, recurse into the left child branch if the value is smaller, or the right if larger
-          (recur (if (< value v) (left n) (right node)))))))
+          (recur (if (< value v) (left n) (right node))) value))))
 
 (def t (utree [5 1 9]))
 (tcontains? t 1)

From 72586375a6a02fa23a5b57b108c75f3ae438e073 Mon Sep 17 00:00:00 2001
From: Paula Gearon <gearon@gmail.com>
Date: Wed, 26 Apr 2023 11:22:00 -0400
Subject: [PATCH 09/12] added missing parameter

---
 doc/scripts/trees.clj | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/doc/scripts/trees.clj b/doc/scripts/trees.clj
index db56e51..1a27d12 100644
--- a/doc/scripts/trees.clj
+++ b/doc/scripts/trees.clj
@@ -441,7 +441,7 @@
            ;; when the node contains the required key, return the associated value
            (value node)
            ;; key not at this node, so look in the children, going left for smaller, right for larger.
-           (recur (if (< c 0) (left node) (right node)))))))
+           (recur (if (< c 0) (left node) (right node)) key)))))
 
 ;; The following map of labels to numbers can be used to demonstrate this function:
 

From 754e4410010c819c58cb153cea76fd9344e4890d Mon Sep 17 00:00:00 2001
From: Paula Gearon <gearon@gmail.com>
Date: Tue, 25 Apr 2023 01:29:02 -0400
Subject: [PATCH 10/12] Added example solutions

---
 doc/scripts/hashsets.clj | 90 ++++++++++++++++++++++++++++++++++++----
 doc/scripts/trees.clj    | 76 +++++++++++++++++++++++++++++----
 doc/scripts/vectors.clj  | 22 +++++++++-
 3 files changed, 173 insertions(+), 15 deletions(-)

diff --git a/doc/scripts/hashsets.clj b/doc/scripts/hashsets.clj
index dfb9825..717138e 100644
--- a/doc/scripts/hashsets.clj
+++ b/doc/scripts/hashsets.clj
@@ -124,7 +124,22 @@ words
 ;; #### Exercise
 ;; Rewrite `insert!` and `contains-object?` to use a linked list at each array location
 
-;; _Solution goes here_
+;; A Solution:
+(defn insert!
+  [arr s]
+  (let [pos (array-location (count arr) s)
+        existing-list (nth arr pos)]
+    (if (some #(= s %) existing-list) ;; search the list
+      arr  ;; already there, so just return the original array
+      (aset arr pos (cons s existing-list))))) ;; add to the list, and put this back in the array
+
+(defn contains-object?
+  [arr s]
+  ;; find the list at the location in the array
+  (let [found (nth arr (array-location (count arr) s))]
+    ;; search the list
+    (some #(= s %) found)))
+
 
 ;; Let's test this by repopulate the array of words, and adding "apple":
 (def words (object-array 10))
@@ -242,8 +257,10 @@ words50
 
 ;; #### Exercise
 ;; Rewrite `array-location` to use the `hash` function
-
-;; _Solution goes here_
+;; A Solution:
+(defn array-location
+  [size s]
+  (mod (hash s) size))
 
 ;; When using a REPL, the `insert` and `contains-object?` functions will pick up this new version of `array-location`
 ;; (defn insert!
@@ -293,7 +310,29 @@ words50
     (aset p 1 v)
     p))
 
-;; _Solution goes here_
+;; A Solution:
+(defn insert-map!
+  [arr k v]
+  ;; get the location in the array
+  (let [pos (array-location (count arr) k)
+        ;; get the existing data at that location
+        existing (nth arr pos)]
+    ;; check the first of each pair, to see if it matches the key
+    (if-let [[fk fv] (first (filter #(= k (first %)) existing))]
+      ;; see if the value is changing
+      (if (= fv v)
+        arr  ;; no change
+        ;; otherwise, remove the old pair, and add the new one
+        (aset arr pos (cons (pair k v) (remove #(= k (first %)) existing))))
+      ;; the key is not there, so just add the pair
+      (aset arr pos (cons (pair k v) existing)))))
+
+(defn get-map
+  [arr k]
+  ;; find the existing list of pairs
+  (let [found (nth arr (array-location (count arr) k))]
+    ;; search the list for the matching key, then return the value
+    (second (first (filter #(= k (first %)) found)))))
 
 ;; Let's test this out with an array of size 50:
 (def map-array (object-array 50))
@@ -321,7 +360,10 @@ map-array
 ;; Write a `to-seq` function that will convert the hashmap array to a seq of pairs.
 ;; Convert these pairs to 2 element vectors.
 
-;; _Solution goes here_
+;; A Solution:
+(defn to-seq
+  [arr]
+  (mapcat #(map vec %) arr))
 
 ;; Apply this to the map-array:
 (to-seq map-array)
@@ -330,7 +372,16 @@ map-array
 ;; Write a `map-keys` function and a `map-vals` function to duplicate the
 ;; `keys` and `vals` functions for maps:
 
-;; _Solution goes here_
+;; A Solution:
+(defn map-keys
+  [arr]
+  (map first (to-seq arr)))
+
+(defn map-vals
+  [arr]
+  (map second (to-seq arr)))
+
+;; Not all of the exercises have to be hard!
 
 ;; ## Immutability
 ;; In order to follow how to make these structures immutable, let's turn to Tree structures.
@@ -345,7 +396,32 @@ map-array
 ;; Write the functions `insert` and `get-map` to duplicate the previous functionality, but
 ;; now the hashmap is in a vector, and the operations must be immutable.
 
-;; _Solution goes here_
+;; A solution. This includes all operations:
+
+(defn location
+  [size s]
+  (mod (hash s) size))
+
+(defn insert
+  ;; accept a key/value as a pair
+  ([vctr [k v]] (insert vctr k v))
+  ;; accept the key and value as separate arguments
+  ([vctr k v]
+   (let [pos (location (count vctr) k)
+         existing (nth vctr pos)]
+     (if-let [[fk fv] (first (filter #(= k (first %)) existing))]
+       (if (= fv v)
+         vctr
+         (assoc vctr pos (cons [k v] (remove #(= k (first %)) existing))))
+       (assoc vctr pos (cons [k v] existing))))))
+
+(defn get-map
+  [vctr k]
+  (let [found (nth vctr (location (count vctr) k))]
+    (second (first (filter #(= k (first %)) found)))))
+
+;; Neither the `location` not the `get-map` function needed to change.
+;; only the `insert` function changed, to use `assoc` rather than `aset`
 
 ;; Initialize a vector at the full width, full of nils:
 (def vctr (vec (repeat 50 nil)))
diff --git a/doc/scripts/trees.clj b/doc/scripts/trees.clj
index d492a5a..986a761 100644
--- a/doc/scripts/trees.clj
+++ b/doc/scripts/trees.clj
@@ -93,7 +93,30 @@
 
 [[5, [1, nil, [3, nil, nil]], [9, nil, nil]]]
 
-;; _Solution goes here_
+
+;; A Solution:
+(defn put!
+  [tree value]
+  (if-not (root tree)
+    ;; if empty, create a new tree with just one node
+    (set-root! tree (node value))
+    ;; iterate down the branches
+    (loop [current-node (root tree)]
+      ;; read the number at the current node
+      (let [current-value (data current-node)]
+        (if (= current-value value)
+          ;; when equal to the number being inserted, then do nothing and return.
+          tree
+          ;; determine the side. Smaller goes to the left, larger to the right.
+          (let [side (if (< value current-value) :left :right)]
+            (if-let [child-node (child current-node side)]
+              ;; if a child node exists on that side, step down and repeat
+              (recur child-node)
+              ;; no child on the selected side, so create a leaf node and insert it
+              (do
+                (set-child! current-node side (node value))
+                ;; return the root of the tree
+                tree))))))))
 
 ;; Try it out...
 
@@ -114,7 +137,17 @@
 ;; Calling `(t-seq t)` should return `(1 3 5 9)`
 
 
-;; _Solution goes here_
+;; A Solution:
+(defn as-seq
+  [node]
+  (and node
+       (concat (as-seq (left node))
+               [(data node)]
+               (as-seq (right node))))) 
+
+(defn t-seq
+  [tree]
+  (as-seq (root tree))) 
 
 (t-seq t)
 
@@ -177,13 +210,24 @@
 ;; #### Exercise:
 ;; Write a function `utree` that creates a tree from a seq parameter.
 
-;; _Solution goes here_
+;; A solution:
+(defn utree [s] (reduce put nil s))
 
 ;; #### Exercise:
 ;; Write a function called `tcontains?` which checks if a value has been stored in the tree. This is the same operation
 ;; as `clojure.core/contains?`
 
-;; _Solution goes here_
+;; A solution:
+(defn tcontains?
+  [node value]
+  (and node   ;; test if the tree rooted at `node` is not empty
+       ;; look at the data for the current node
+       (let [v (data node)]
+         (or
+           ;; if the data node data is the same as what is being looked for, then return true
+          (= value v)
+          ;; otherwise, recurse into the left child branch if the value is smaller, or the right if larger
+          (recur (if (< value v) (left n) (right node)))))))
 
 (def t (utree [5 1 9]))
 (tcontains? t 1)
@@ -315,7 +359,12 @@
 ;; #### Exercise:
 ;; Write a `max-depth` function to return the maximum depth of the rb-tree. Hint: use recursion.
 
-;; _Solution goes here_
+;; A Solution:
+(defn max-depth
+  [n]
+  (if n
+    (inc (max (max-depth (left n)) (max-depth (right n))))
+    0))
 
 ;; The maximum depth can indicate that a tree is relatively balanced. A perfectly balanced tree
 ;; ought to be about the log2 of the number of items. Red/Black trees do not balance perfectly, but we should
@@ -381,7 +430,18 @@
 ;; Write `tget`: a function that adds to the `tcontains?` function to search for a given key,
 ;; returning the associated value.
 
-;; _Solution goes here_
+;; A solution:
+(defn tget
+  [node key]
+  ;; return nil for an empty tree
+  (and node
+       ;; compare the value at this node to the key being searched for
+       (let [c (compare key (nkey node))]
+         (if (zero? c)
+           ;; when the node contains the required key, return the associated value
+           (value node)
+           ;; key not at this node, so look in the children, going left for smaller, right for larger.
+           (recur (if (< c 0) (left node) (right node)))))))
 
 ;; The following map of labels to numbers can be used to demonstrate this function:
 
@@ -403,7 +463,9 @@
 ;; Implement the `tkeys` and `tvals` functions that return the keys and values of this tree set as seqs.
 ;; Hint: you can use the `as-seq` from above.
 
-;; _Solution goes here_
+;; A Solution:
+(defn tkeys [tree] (map first (as-seq tree)))
+(defn tvals [tree] (map second (as-seq tree)))
 
 (tkeys numbers)
 (tvals numbers)
diff --git a/doc/scripts/vectors.clj b/doc/scripts/vectors.clj
index 5b0691c..5c7b29f 100644
--- a/doc/scripts/vectors.clj
+++ b/doc/scripts/vectors.clj
@@ -152,7 +152,27 @@ v
 ;; Write a function called vth that can manually retrieve the nth element from a vector that contains
 ;; between 1057 and 32800 elements. i.e. There are 3 levels.
 
-;; _Solution goes here_
+;; A Solution:
+(defn vth
+  [v n]
+  ;; check that the offset is not out of range
+  (when (< n (count v))
+    ;; get the offset in the root
+    (let [first-level (int (/ n 1024))
+          ;; record the remaining offset
+          n2 (mod n 1024)
+          ;; the offset in the second level of the tree
+          second-level (int (/ n2 32))
+          ;; the final offset
+          third-level (mod n2 32)]
+      ;; check if the offset is beyond the range of the first level
+      (if (or (> first-level 32)
+              ;; or the second level offset is a nil node
+              (nil? (-> v .root .array (nth first-level) .array (nth second-level))))
+        ;; this indicates that the offset is beyond the current data and inside the tail
+        (-> v .tail (nth third-level))
+        ;; otherwise, traverse down the tree to the offset
+        (-> v .root .array (nth first-level) .array (nth second-level) .array (nth third-level))))))
 
 (vth vec1057 500)
 ;; 500

From 60d310113727a4c21408cd00c2fbb2bd83f50db7 Mon Sep 17 00:00:00 2001
From: Paula Gearon <gearon@gmail.com>
Date: Wed, 26 Apr 2023 11:01:09 -0400
Subject: [PATCH 11/12] added missing param

---
 doc/scripts/trees.clj | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/doc/scripts/trees.clj b/doc/scripts/trees.clj
index 986a761..db56e51 100644
--- a/doc/scripts/trees.clj
+++ b/doc/scripts/trees.clj
@@ -227,7 +227,7 @@
            ;; if the data node data is the same as what is being looked for, then return true
           (= value v)
           ;; otherwise, recurse into the left child branch if the value is smaller, or the right if larger
-          (recur (if (< value v) (left n) (right node)))))))
+          (recur (if (< value v) (left n) (right node))) value))))
 
 (def t (utree [5 1 9]))
 (tcontains? t 1)

From 773065c11175ab974e73258e247710ade0e50d42 Mon Sep 17 00:00:00 2001
From: Paula Gearon <gearon@gmail.com>
Date: Wed, 26 Apr 2023 11:22:00 -0400
Subject: [PATCH 12/12] added missing parameter

---
 doc/scripts/trees.clj | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/doc/scripts/trees.clj b/doc/scripts/trees.clj
index db56e51..1a27d12 100644
--- a/doc/scripts/trees.clj
+++ b/doc/scripts/trees.clj
@@ -441,7 +441,7 @@
            ;; when the node contains the required key, return the associated value
            (value node)
            ;; key not at this node, so look in the children, going left for smaller, right for larger.
-           (recur (if (< c 0) (left node) (right node)))))))
+           (recur (if (< c 0) (left node) (right node)) key)))))
 
 ;; The following map of labels to numbers can be used to demonstrate this function: