A073345(n,k) is number of rooted binary trees of size n and height k.

A073345(n,n) is number of rooted binary trees of size n and height n.
Inductive hypothesis A073345(n,n) = 2^(n-1) for n>0:
A073345(1,1) = 1, 
and A073345(n+1,n+1)=2*A073345(n,n) since for each rooted binary tree of size n and height n there are only two places to put the extra node which increases the height by 1 and each new rooted binary tree is unique, 
so hypothesis true by induction.

A073345(n,n-1) = (n-2)*A073345(n-1,n-1)-(1/2)*A073345(n-1,n-1) for n>2 since for each rooted binary tree of size (n-1) and height (n-1) there are (n-2) places to put the extra node which does not increase the height but not all of these each new rooted binary tree are unique (when the extra node is put as high as possible, then a set of pairs of identical new trees is produced),
so A073345(n,n-1) = (n-2)*2^(n-2)-(1/2)*2^(n-2) = (2n-5)*2^(n-3) = (2*(n-3)+1)*2^(n-3) = A014480(n-3).

A073346(n,n-1) = A073345(n,n) = 2^(n-1) for n>0 since the only rooted binary trees of size n and "contracted height" n-1 are rooted binary trees of size n and height n with the top node contracted by 1, and there is a direct correspondence between them.

A073346(n,n-2) = A073345(n,n-1)-A073345(n-2,n-2) for n>2 since the only rooted binary trees of size n and "contracted height" n-2 are rooted binary trees of size n and height n-1 with the top node contracted by 1, but there is not a direct correspondence between them (when a rooted binary trees of size n and height n-1 has to be contracted by 2 at the top, corresponding to a rooted binary tree of size (n-2) and height (n-2) with the top node directly attached to two extra nodes above it),
so A073346(n,n-2) = (2n-5)*2^(n-3)-2^(n-3) = (2n-6)*2^(n-3) = (n-3)*2^(n-2) = A058922(n-2).