In answering a question involving tree counting, I ran across the Motzkin numbers, which I had never encountered before. The problem involved counting the number of rooted, ordered trees with $n$ nodes, subject to the condition that no nodes has degree greater than 3. Basically, I solved the problem by finding a recurrence, computing some terms, and sticking the result in OEIS. This turned up the Motzkin numbers, except that I had $T_n=M_{n-1}.$ (Now that I think of it, the correspondence would have been perfect if I'd changed to counting trees with $n$ edges.) Except for the difference in index, the two sequences had the same initial values, and satisfied the same recurrence, so they are identical.
Anyway, I see on Wikipedia that one way to define $M_n$ is
the Motzkin number for $n$ gives the number of routes on the upper right quadrant of a grid from coordinate $(0, 0)$ to coordinate $(n, 0)$ in $n$ steps if one is allowed to move only to the right (up, down or straight) at each step but forbidden from dipping below the $y = 0$ axis.
I have been trying to find a bijective proof that the number of such paths equals the number of admissible trees with $n$ edges, but I'm not having any luck. I can see that going up corresponds to a node with two children, going straight to an node with one child, and going down to a node with no child, but I can't figure out how to make the correspondence in detail.
I've drawn the $9$ trees with $4$ edges, and I've been comparing them to the paths pictured on Wikipedia, and I can match up some of them, but not all. For example, there are only two trees that have two nodes of degree $3$, and there are only two paths shown that go up twice, so the pairs must correspond. However, the two trees are symmetric, in that they correspond if left and right are interchanged, and I see no such symmetry in the path.
I've also thought about traversing the tree by depth-first search say, and somehow encoding the edges as Up, Straight, or Down, but I haven't been able to make that work out.
I've tried to find a proof on the Web, but everything I run across seems to be dealing with much more advanced problems.
Please give me a proof, or a hint, or a reference.