What is the number of full binary trees of height less than $h$ Given a integer $h$

What is $N(h)$ the number of  full binary trees of height less than $h$?


For example $N(0)=1,N(1)=2,N(2)=5, N(3)=21$(As pointed by TravisJ in his partial answer) I can't find any expression of $N(h)$ neither a reasonable upper bound.
Edit In a full binary tree (sometimes called proper binary tree) every node other than the leaves has two children.

You can define $T(h,b)$ as the number of trees of height $h$ with $b$ leaves at the bottom.  Your $N(h)$ is then the sum over $b$ of $T(h,b)$  To get a tree of height $h+1$ and $b$ leaves on the bottom, you pick a tree of height $h$ and at least $b/2$ leaves on the bottom, then pick $b/2$ leaves to hang new leaves from.  This gives a recurrence $$T(h+1,b)=\sum_{k=b/2}^{2^h}T(h,k){k \choose b/2}$$
We have $T(3,2)=8, T(3,4)=8, T(3,6)=4, T(3,8)=1$, giving the total of $21$ cited by TravisJ.    Then $T(4,2)=8 \cdot 2 + 8 \cdot 4 + 4 \cdot 6 + 1 \cdot 8=80, T(4,4)=8\cdot 1+8\cdot 6 + 4 \cdot 15+1\cdot 28=144, T(4,6)=8\cdot 4+4 \cdot 20+1\cdot 56=168,T(4,8)=8\cdot 1+4\cdot 15+1\cdot 70=138,T(4,10)=4\cdot 6+1\cdot 56=80,T(4,12)=4\cdot 1+1\cdot 28=32,T(4,14)=8,T(4,16)=1$ for a total of $N(4)=651$ The sequence is given in OEIS A001699 where it is said to approach $1.5028368...^{(2^n)}$ and some references are given.
