In this paper, we focus on so-called
*basketball walks*, which are integer-valued walks with step-set {-2,-1,+1,+2}.
We give an explicit bijection that maps, for each *n* >= 2, *n*-step
basketball walks from 0 to 0 that visit 1 and are positive
except at their extremities to *n*-leaf binary trees. Moreover, we can
partition the steps of a walk into +-1-steps, odd +2-steps or
even -2-steps, and odd -2-steps or even +2-steps, and these
three types of steps are mapped through our bijection to double
leaves, left leaves, and right leaves of the corresponding tree.

We also prove that basketball walks from 0 to 1 that are positive
except at the origin are in bijection with increasing unary-binary
trees with associated permutation avoiding 213. We furthermore give
the refined generating function of these objects with an extra
variable accounting for the unary nodes.

Received: November 5, 2016. Accepted: December 2, 2016. Final version: January 19, 2017.

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