TL;DR

On with TASK #2 from The Weekly Challenge #130. Enjoy!

# The challenge

You are given a tree.

Write a script to find out if the given tree is Binary Search Tree (BST).

According to wikipedia, the definition of BST:

A binary search tree is a rooted binary tree, whose internal nodes each store a key (and optionally, an associated value), and each has two distinguished sub-trees, commonly denoted left and right. The tree additionally satisfies the binary search property: the key in each node is greater than or equal to any key stored in the left sub-tree, and less than or equal to any key stored in the right sub-tree. The leaves (final nodes) of the tree contain no key and have no structure to distinguish them from one another.

Example 1

Input:
8
/ \
5   9
/ \
4   6

Output: 1 as the given tree is a BST.


Example 2

Input:
5
/ \
4   7
/ \
3   6

Output: 0 as the given tree is a not BST.


# The questions

My question about the definition is whether the leaves are considered only the empty left/right nodes of a node that has a key. Another question would be whether a node with a key always has two non-empty left and right children.

All in all, anyway, it doesn’t really matter for the implementation I have in mind… so it’s more curiosity than anything else.

# The solution

The most straightforward approach is, for me, to go recursive. In this case, in each node we will have to consider the following quantities:

• the key of the node itself;
• the minimum key and the maximum key of the left child, which we will call $lmin and $lmax;
• the same quantities for the right child, respectively $rmin and $rmax.

At that node, we have the following:

• if either the left or the right child don’t comply with the BST rules, then the whole tree does not either. Hence, we have to make sure that they do.
• At the specific node, we must check that the key is greater than $lmax (i.e. the maximum value on the left side) and that it is also less than $rmin (i.e. the minimum value on the right side).

If both apply, then this particular node is good, and we can go back to the parent node, reporting a success and also $lmin and $rmax as the overall minimum and maximum values.

In case the tree is not perfectly assembled (e.g. a node only has a left or a right side) we will have to cope with the fact and act accordingly.

Perl goes first this time:

#!/usr/bin/env perl
use v5.24;
use warnings;
use experimental 'signatures';
no warnings 'experimental::signatures';

sub check_bst ($root) { state$checker = sub ($node) { return 1 unless$node;
my $key =$node->{key};
my ($lsub,$lmin, $lmax) = __SUB__->($node->{left});
return 0 unless $lsub; ($lmin, $lmax) = ($key, $key - 1) unless defined$lmin;
my ($rsub,$rmin, $rmax) = __SUB__->($node->{right});
return 0 unless $rsub; ($rmin, $rmax) = ($key + 1, $key) unless defined$rmin;
return 0 if $key <$lmax || $key >$rmin;
return (1, $lmin,$rmax);
};
return ($checker->($root));
}

sub n ($k,$l = undef, $r = undef) {{key =>$k, left => $l, right =>$r}}

say check_bst(n(8, n(5, n(4), n(6)), n(9)));
say check_bst(n(5, n(4, n(3), n(6)), n(7)));


I guess that these two lines deserve some additional explanation:

($lmin,$lmax) = ($key,$key - 1) unless defined $lmin; ... ($rmin, $rmax) = ($key + 1, $key) unless defined$rmin;


In case a leg is empty, we get nothing from it (i.e. undef). This inherently means that the sub-tree on that side is compliant, hence:

• to make the test succeed, we set $lmax to be smaller than the key ($key - 1), and $rmin to be greater than it ($key + 1).
• on the other hand, the missing extreme is set to be equal to $key, because this is the value we want to send back to the parent’s call. Time for Raku now, which is a simple translation: #!/usr/bin/env raku use v6; sub check-bst ($root) {
my sub checker ($node --> Array()) { return 1 unless$node;
my ($key,$left, $right) =$node<key left right>;
my ($lsub,$lmin, $lmax) = checker($left);
return 0 unless $lsub; ($lmin, $lmax) = ($key, $key - 1) unless defined$lmin;
my ($rsub,$rmin, $rmax) = checker($right);
return 0 unless $rsub; ($rmin, $rmax) = ($key + 1, $key) unless defined$rmin;
return 0 if $key <$lmax || $key >$rmin;
return (1, $lmin,$rmax);
}
return checker($root); } sub n ($k, $l = Nil,$r = Nil) {(key => $k, left =>$l, right => \$r).hash}

put check-bst(n(8, n(5, n(4), n(6)), n(9)));
put check-bst(n(5, n(4, n(3), n(6)), n(7)));


And with this… it’s all for this post, I hope you enjoyed it and stay safe anyway!