本篇論文討論了二次代數生成函數 (quadratic generating function) 及相關的組合結構，並得出一些這個領域上的新結果。第二章整理了具有二次代數生成函數的特殊二次組合數及組合結構。第三章提出處理二次代數生成函數的Taylor 展開式法，據此解決Motzkin path with flaws的計數問題。第四章中證明，由一個二次代數生成函數可誘導出一對滿足二次三項遞迴式的多項式族。第五章解決Schröder path with flaws的計數。第六章解決了任意給定山峰高度的Dyck路徑的計數問題，並引進等價類的概念。第七章利用對應方法給出了關於根樹以及網著色的組合證明。第八章解決非自交非孤立點分割的區塊計數。

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