How to make the "finals tree table", or what is it called?

I'm working with some friends on a small private mini MMORPG game server, and we want to make a PvP event (players VS player) fights, and post the live results on the website as a tree.

I don't know what is it called, I tried every keyword combination on google and this is what I finaly came up with (2 pictures at the bottom).

I need the name of the two types of the trees, what is called so I can do more googling about it.

I also need to know if there is any PHP/HTML/JS or FLASH, Scripts or API or something that can help me accomplish this. I have no problem coding it from PHP/MySQL side, but it looks very hard to render and print on client side, because the tree can get really big if we get alot of players. Please help.

enter image description here

Solution

This is called a complete full binary tree I think.

Here you can find some mathematical info to binary-trees.