# Find the longest possible path in full binary tree

Given the depth of the tree, I need to calculate the longest possible path in the full binary tree (also known as the diameter).

When attempting this problem, I experimented with what the depth has to do with the path length. However, the solutions I came up with depended on going through the root, while the longest path could obviously exist outside of the root.

What's an equation I can use to find the diameter of the full binary tree using the depth?

• If "full binary tree" you mean "fully occupied" so it is balanced you can use height. Otherwise traversing the tree will be needed. – Evil Dec 11 '15 at 7:57

Then you know the depth $$D$$ will be half of the total possible diameter. This is because we can take a maximum possible path of length $$D$$ from root to any leaf in the subtree rooted at the left-child of the root, and we can also take a maximum possible path of length $$D$$ from root to any leaf in the subtree rooted at the right-child of the root. Thus, adding these up would be a path of length $$2D$$.
Thus, we get that the maximum possible diameter would be equal to twice the depth (i.e. $$\mathrm{diameter} = 2\cdot \mathrm{depth}$$).
• @HendrikJan that is a good point. It is not necessarily the case. However $2D$ would still be the longest possible path, so I think (?) it still answers the question unless OP would clarify what they mean by full. – ryan Apr 10 at 20:42