AVL tree with fixed height and as few elements as possible

I have been reading about AVL trees, at the moment I'm trying to figure out how to determine the height of a tree and how to draw an AVL tree of some height with minimum number of elements.

In a tutorial I found that this: would be a AVL tree of height 7

And this AVL tree with the height 4

This is really confusing by the look I would guess that both of them are of height 4. I'm fairly new to data structures, I could not find a simple documentation/tutorial regarding this most of what i found was about Insertion/Deletion with AVL trees.

So is the top tree of height 7 if not how would I draw it with the minimal number of elements. I understand the each sub tree would have to be balanced.

Generally tree height is defined as the length of the longest path from the root to a leaf. Therefore you're right that the two trees have the same height, but they actually have height 3.

As far as your second question, about how to draw the smallest possible AVL tree of a given height $n$, the trick is to think inductively. Construct a new tree of height $n$ by joining the smallest possible AVL trees of height $n-1$ and height $n-2$ as subtrees of a new common root. Using the definition of AVL trees, it's a straightforward exercise to see that (1) this new tree is a valid AVL tree and (2) it's the smallest possible AVL tree of height $n$.

To get the height 7 tree in particular, find the base case minimal AVL trees of heights 0 and 1, and repeat the above construction 6 times.

StackExchange Computer Science Q#11086, answer score: 4