If I store a binary tree in an array, how do I avoid the wasted space?

Question

Often we need trees in algorithms and I get out a tree with lots of pointers and recursion.
Sometimes I need more speed an I put the tree into an 2D array like so:

Example of a binary tree stored in an array
+-----------------+
|0eeeeeeeeeeeeeeee| //no pointers needed, parent/child, is y dimension,
|11       dddddddd| //sibbling is x dimension of the array.
|2222         cccc|  //The 123 tree is stored root up.
|33333333       bb|  //Notice how the abc-tree is stored upside down 
|4444444444444444a|  //The wasted space in the middle, is offset by the fact 
+-----------------+  //that you do not need up, down, and sibbling pointers.

I love this structure because it allowes me speed up options that I don't have when using pointers and recursion.

But notice that wasted space in the middle....

How do I get rid of/reuse that wasted space?

Requirements
I only use this structure if I need every last bit of speed, so a solution with lots of translations and address calculations to get to that space will not be helpful.

Answer 1

A binary tree can be stored in an array more efficiently as explained here: http://en.wikipedia.org/wiki/Binary_tree#Arrays:

From Wikipedia:

In this compact arrangement, if a node has an index i, its children are found at indices (2i + 1) (for the left child) and (2i + 2) (for the right), while its parent (if any) is found at index floor((i-1)/2) (assuming the root has index zero).

This method benefits from more compact storage and better locality of reference, particularly during a preorder traversal. However, it is expensive to grow and wastes space proportional to (2H - n) for a tree of height H with n nodes.

In other words, it will waste space if your tree is not a complete binary tree, but it will still be a bit more compact than a plain 2D array.