棋

Left tree radical - chess pieces and boards have been wooden across Chinese history. Anchors 棋 firmly in the wooden-object family: 椅 chair, 桌 table, 桶 bucket, 棒 stick. The radical declares the material immediately. Same family also gives 板 (board) - close cousin since chessboards are 棋板.

Right 其 supplies the sound - qí unchanged. 其 itself originally pictured a winnowing basket; later borrowed for the classical pronoun 'his/its'. Pure phonetic role. Same phonetic family: 期 (period), 基 (foundation), 旗 (flag), 欺 (deceive) - tight qí / jī / qī cluster. 围棋 (Go), 象棋 (Chinese chess), 国际象棋 (Western chess) all use 棋 as the core.