The OEIS entry [A121448][1] enumerates binary trees with $n$ edges and $k$ vertices with outdegree 1. 

Has anyone seen these trees?

The o.g.f. for this entry, $G(x,t)$, is essentially a discriminant whose associated row polynomials can be generated as an umbral composition of the reverse face polynomials of the hypercubes [A038207][2] with the aerated Catalan numbers [A126120][3]. The compositional inverse of $G$ is the o.g.f. of the [Chebyshev polynomials][4] of the second kind [A053117][5] (mod signs). More generally, the row polynomials of G are invariant polynomials of the compositional inverses of the o.g.f.s (times a factor $x^{\alpha}$) of the [Gegenbauer polynomials][6] of differing order $\alpha$. The row polynomials of $G$ are also related to the Motzkin paths of [A097610][7]. 

Given the rich associations among these combinatoric structures, polygon dissections, lattice paths, orthogonal polynomials, Cartan-Lie algebras, and compositional inverses and algebraic geometry/topology, I would hope these binary trees (or row polynomials) have explicitly popped up in other literature.


  [1]: http://oeis.org/A121448
  [2]: http://oeis.org/A038207
  [3]: http://oeis.org/A126120
  [4]: http://en.wikipedia.org/wiki/Chebyshev_polynomials
  [5]: http://oeis.org/A053117
  [6]: http://en.wikipedia.org/wiki/Gegenbauer_polynomials
  [7]: http://oeis.org/A097610