Trees

BST Insert

Insert values into a binary search tree by comparing at each node.

Algorithm

The canonical tree is 4(2(1,3),6(5,7)), so this Perl DSA implementation can be compared directly with the rest of the DSA track.

binary search tree Values smaller than a node go left; larger values go right.

Visual walkthrough

BST insertion is a comparison path. The pinned tree 4(2(1,3),6(5,7)) is shown with the inserted value taking its sorted slot.

Step 1 - Start at root

For value 5, compare with 4 first; 5 is larger, so move right.

First comparison: 5 > 4, so the search for the insert slot goes right.insert 54compare26137

Step 2 - Take the left slot under 6

At 6, value 5 is smaller, so it becomes the left child.

Second comparison: 5 < 6, so the open left slot is used.426compare135new7

Step 3 - Canonical tree

The resulting tree is the pinned shape 4(2(1,3),6(5,7)).

Final BST after 5 is present under 6.4261357

Basic Implementation

basic.pl 
use strict;
use warnings;
sub node { my ($value, $left, $right) = @_; return { value => $value, left => $left, right => $right }; }
sub render {
    my ($node) = @_;
    return "_" unless defined $node;
    return "$node->{value}" unless defined $node->{left} || defined $node->{right};
    return "$node->{value}(" . render($node->{left}) . "," . render($node->{right}) . ")";
}
sub sample_tree {
    return node(4, node(2, node(1), node(3)), node(6, node(5), node(7)));
}
sub list_string { return "[" . join(", ", @_) . "]"; }
sub insert { my ($root, $value) = @_; return node($value) unless defined $root; if ($value < $root->{value}) { $root->{left} = insert($root->{left}, $value); } else { $root->{right} = insert($root->{right}, $value); } return $root; }
my $root; for my $value (4, 2, 6, 1, 3, 5, 7) { $root = insert($root, $value); } print render($root) . "\n";

Complexity

  • Time: O(h) per insert
  • Space: O(n)

Implementation notes

  • node($value, $left, $right) returns a Perl hash reference: { value => $value, left => $left, right => $right }.
  • Missing children are undef; render prints an undefined child as _.
  • Nodes are accessed with hash-reference syntax such as $root->{value} and $root->{left}.
  • my $root starts undefined, then each inserted value reassigns $root = insert($root, $value).
  • insert takes ($root, $value) and returns node($value) when it reaches an undefined subtree.
  • The comparison is numeric: if ($value < $root->{value}).
  • Smaller values recurse left with $root->{left} = insert($root->{left}, $value).
  • All other values recurse right with $root->{right} = insert($root->{right}, $value), so duplicates would follow the right branch in this source.
  • The checked insert order is 4, 2, 6, 1, 3, 5, 7.
  • The trace builds root 4, then attaches 2 on the left and 6 on the right, producing 4(2,6).
  • Inserting 1 follows 4 -> left -> 2 -> left; inserting 3 follows 4 -> left -> 2 -> right, giving 4(2(1,3),6).
  • Inserting 5 follows 4 -> right -> 6 -> left; inserting 7 follows 4 -> right -> 6 -> right, producing 4(2(1,3),6(5,7)).
  • print render($root) . "\n" prints that final tree string instead of dumping Perl object or hash-reference internals.
  • The replay also includes a sorted-insert contrast, 1(_,2(_,3(_,4))), to show the unbalanced height-4 shape without rotations.