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## Some balanced binary trees in APL

A few articles have been written on basic tree wrangling in APL. Here I try to aggregate them and use them for balanced binary trees. In general I do not recommend working with trees in APL, but if push comes to shove you can use this, or mess with Dyalog’s .NET integration for a reference-based version.

Let’s take the following tree as example:

``````27 (1)
+ - 14 (2)
|   + - 10 (3)
|   |   + - 3 (4)
|   + - 19 (5)
+ - 35 (6)
+ - 31 (7)
+ - 42 (8)
``````

#### BST (Binary Search Tree)

Aaron Hsu repr:

Let’s call this AH.

Uses a parent vector, left sibling, data vectors.

We use position here rather than node to the left because position matters in a binary search tree.

``````ap←1  1  2  3 2  1  6  6 ⍝ parent nodes
al←1  1  1  1 2  1  1  1 ⍝ position (left=1/right=2)
ad←27 14 10 3 19 35 31 42 ⍝ data
``````

John Earnest’s pair-based repr:

Let’s call this JE. The second version explored on the page is very similar to Aaron Hsu’s idea.

``````jt←↑(2 6) (3 5) (4 0) (0 0) (0 0) (7 8) (0 0) (0 0) ⍝ child nodes
jd← 27    14    10    3     19    35    31    42    ⍝ data
``````
##### Searching and Insertion

Effectively a binary search which is arguably better done by sorting the data array and using interval index. However, it is not too complex to do a search on both tree representations:

For the JE repr, since the number of nodes is part of the structure, it is somewhat simple to modify the data:

``````FindJE ← {
(t d)←⍺
s←⍵
1∘{x←⍺⊃d ⋄ n←(1+⍵>x)⊃⍺⌷t ⋄ (⍵=x)∨0=n:⍺ ⋄ n∇⍵}s
}
InsertJE ← {
(t d)←⍺
s←⍵
v←⍺ FindJE ⍵
t←{(1+≢d)@(⊂v(1+s>v⊃d))⊢⍵}⍣(0≠≢d)⊢t⍪1
t (d,⍵)
}
``````

For AH, I found it much simpler to use a recursive function to find a required node.

``````FindAH ← {
(p l d)←⍺ ⍝ full tree
s←⍵ ⍝
1∘{x←⍺⊃d ⋄ (⍵=x)∨~(1+⍵>x)∊(⍺=p)/l:⍺ ⋄ ((1+⍵>x)⊃1~⍨⍸⍺=p)∇⍵}s
}
InsertAH ← {
(p l d)←⍺
⎕←v←⍺ FindAH ⍵
(p,v) (l,1+⍵>v⊃d) (d,⍵)
}
``````

#### AVL trees

Here we will be using Aaron Hsu’s repr only, for an AVL tree, we can track depths of each vertex:

``````d ←  27    14    10    19    35    31    42
dp ← 1     2     3     3     2     3     3
``````
##### Rotation

The steps for left rotation about a given node are:

so, going by this, we can define a right rotation for AH about a node given by index `y` as follows:

1. x and y swap parents
2. x gets y’s position (left or right)
3. T2 becomes left node of y.
4. update heights of y and x.

Here, we don’t need to change anything except for the parent and height of each node.

``````RRotAH ← {
(p l d h)←⍺
y←⍵
x←⊃⍸(l=1)∧p=y
p[x]←p[y]          ⍝ Step 1
p[y]←x
l[x]←l[y] ⋄ l[y]←2 ⍝ Step 2
t2←⊃⍸(l=2)∧p=x
p[t2]←y ⋄ l[t2]←1  ⍝ Step 3
h[y]←1+⌈/h[⍸p=y]   ⍝ always update y first, it is the child node.
h[x]←1+⌈/h[⍸p=x]   ⍝ Step 4
p l d h
}
``````

A similar technique can be used for a left rotation:

``````LRotAH ← {
(p l d h)←⍺
x←⍵
⎕←'y: ',y←⊃⍸(l=2)∧p=x
t2←⊃⍸(l=1)∧p=y
p[y]←p[x]          ⍝ Step 1
p[x]←y
l[y]←l[x] ⋄ l[x]←1 ⍝ Step 2
p[t2]←x ⋄ l[t2]←2  ⍝ Step 3
h[x]←1+⌈/h[⍸p=x]   ⍝ always update y first, it is the child node.
h[y]←1+⌈/h[⍸p=y]   ⍝ Step 4
p l d h
}
``````

Finally, we can insert stuff in the tree with the existing BST insertion, and these rotations. There’s four cases listed at geeksforgeeks, which are somewhat simple to handle with indexing.

Code for these is available in this gist.