Figure 1: Chess openings can be described as decision trees, showing each move and associated branching ratios. This diagram shows the three most popular first $(d=1)$ half-moves in the $1.5$-million-game ScidBase chess database [12] and their branching ratios. For example, in $45\%$ of the games, white starts with e4 (King’s pawn to fourth row, in algebraic chess notation), $35\%$ start with d4 (Queen’s pawn to fourth row), etc. Each of these moves then have branching ratios to the second half-move by black $(d=2)$. Blasius and Tönjes find that for all games up to $d=40$, the opening sequence popularity follows a Zipf power law with universal exponent nearly equal to $-2$, but for small values of $d$, the exponent is nonuniversal and depends linearly on $d$. (Adapted from Ref. [1].)