Non-Contiguous Pattern Avoidance in Binary Trees
  In this paper we consider the enumeration of binary trees avoiding non-contiguous binary tree patterns. We begin by computing closed formulas for the number of trees avoiding a single binary tree pattern with 4 or fewer leaves and compare these results to analogous work for contiguous tree patterns. Next, we give an explicit generating function that counts binary trees avoiding a single non-contiguous tree pattern according to number of leaves. In addition, we enumerate binary trees that simultaneously avoid more than one tree pattern. Finally, we explore connections between pattern-avoiding trees and pattern-avoiding permutations. 
Michael Dairyko, Lara Pudwell, Samantha Tyner, and Casey Wynn
math.CO