You could always, of course, use forest:

\documentclass[tikz]{standalone}
\usepackage{forest}
\usepackage{skak}
\usepackage{chessboard}
\let\boardfont\undefinedcommand
\usepackage{diagram}
\begin{document}
\setboolean{showcomputer}{false}
\setboolean{piececounter}{false}
\specialdiagnum{}
\begin{forest}
for tree={
edge path={
\noexpand\path[\forestoption{edge}, shorten <=-15pt, line width=2.5pt] (!u.parent anchor) -- (.child anchor)\forestoption{edge label};
},
}
[{\begin{diagram}[4x4]\end{diagram}}
[{\begin{diagram}[4x4]\pieces{wDa4}\end{diagram}}
[{\begin{diagram}[4x4]\pieces{wDa4, wDc3}\end{diagram}}]
[{\begin{diagram}[4x4]\pieces{wDa4, wDd3}\end{diagram}}
[{\begin{diagram}[4x4]\pieces{wDa4, wDd3, wDb2}\end{diagram}}]
]
]
[{\begin{diagram}[4x4]\pieces{wDb4}\end{diagram}}
[{\begin{diagram}[4x4]\pieces{wDb4, wDd3}\end{diagram}}
[{\begin{diagram}[4x4]\pieces{wDb4, wDd3, wDa2}\end{diagram}}
[{\begin{diagram}[4x4]\pieces{wDb4, wDd3, wDa2, wDc1}\end{diagram}}]
]
]
]
[{\begin{diagram}[4x4]\pieces{wDc4}\end{diagram}}
[{\begin{diagram}[4x4]\pieces{wDc4, wDa3}\end{diagram}}
[{\begin{diagram}[4x4]\pieces{wDc4, wDa3, wDd2}\end{diagram}}
[{\begin{diagram}[4x4]\pieces{wDc4, wDa3, wDd2, wDb1}\end{diagram}}]
]
]
]
[{\begin{diagram}[4x4]\pieces{wDd4}\end{diagram}}
[{\begin{diagram}[4x4]\pieces{wDd4, wDa3}\end{diagram}}
[{\begin{diagram}[4x4]\pieces{wDd4, wDa3, wDc2}\end{diagram}}]
]
[{\begin{diagram}[4x4]\pieces{wDd4, wDb3}\end{diagram}}]
]
]
\end{forest}