Almost complete binary tree of height
h
For
h
= 0, just a single node
For
h
= 1, left child or two children
For
h
≥ 2, either:
the left subtree of the root is
complete
with height
h
-1 and the right is
almost complete
with height
h
-1, OR
the left is
almost complete
with height
h
-1 and the right is
complete
with height
h
-2
push() expected running time
How far to move up?
Half of the nodes are leaves, so half of the inserts will only move up one level
A quarter of the nodes are one level above the leaves, so one quarter of the inserts will move up two levels
One eighth will require moving up 3 levels
One sixteenth will require moving up 4 levels
Expected running time:
1/2 * 1 + 1/4 * 2 + 1/8 * 3 + ... = Sum((1/(2^i))*i = 2
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