On Chip-Firing on Undirected Binary Trees

Abstract Chip-firing is a combinatorial game played on an undirected graph in which we place chips on vertices and disperse them. We study chip-firing on an infinite binary tree in which we add a self-loop to the root to ensure each vertex has degree 3. A vertex can fire if the number of chips placed on it is at least its degree. In our case, a vertex can fire if it has at least three chips, and it fires by dispersing one chip to each neighbor. Motivated by a 2023 paper by Musiker and Nguyen on this setting of chip-firing, we give an upper bound for the number of stable configurations when we place $$2^\ell - 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>ℓ</mml:mi> </mml:msup> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> labeled chips at the root. When starting with N chips at the root where N is a positive integer, we determine the number of times each vertex fires when N is not necessarily of the form $$2^\ell - 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>ℓ</mml:mi> </mml:msup> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . We also calculate the total number of fires in this case.