Build-a-Tree | Phylogenetic tree construction game

What you are trying to do is the same as self-balancing binary search trees but not limited to just max 2 children. You can directly take help from B-Tree or B+ Tree.

B-Tree Insertion says

All insertions start at a leaf node. To insert a new element, search the tree to find the leaf node where the new element should be added. Insert the new element into that node with the following steps:

1. If the node contains fewer than the maximum allowed number of elements, then there is room for the new element. Insert the new element in the node, keeping the node's elements ordered.

2. Otherwise the node is full, evenly split it into two nodes so:

   1. A single median is chosen from among the leaf's elements and the new element.
   2. Values less than the median are put in the new left node and values greater than the median are put in the new right node, with the median acting as a separation value.
   3. The separation value is inserted in the node's parent, which may cause it to be split, and so on. If the node has no parent (i.e., the node was the root), create a new root above this node (increasing the height of the tree).

This shows how to split instead of rotating the trees.

Above excerpt related to B-Tree insertion is kind of a pseudo-code/broad steps in the algorithm. Instead of adding more of pseudo-code, let me take a simulation from here that explains the insert operation. The same link also has the code.

Let us understand the algorithm with an example tree having max 3 children and the same node can hold 5 keys. Consider a sequence of integers 10, 20, 30, 40, 50, 60, 70, 80 and 90 in an initially empty B-Tree.

Initially root is NULL. Let us first insert 10.

Let us now insert 20, 30, 40 and 50. They all will be inserted in root because the maximum number of keys a node can accommodate is is 5.

Let us now insert 60. Since root node is full, it will first split into two, then 60 will be inserted into the appropriate child.

Let us now insert 70 and 80. These new keys will be inserted into the appropriate leaf without any split.

Let us now insert 90. This insertion will cause a split. The middle key will go up to the parent.

I hope this helped!

Feel free to play around with this B-Tree Visualization!

Not to mention, you can always find javascript implementations of B-Tree on the internet e.g. this one.