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Haussler’s convolution kel provides an effective framework for engineering positive semidefinite kels,and has a wide range of applications.On the other hand,the mapping kel that we introduce in this paper is its natural generalization,and will enlarge the range of application significantly.Our main theorem with respect to positive semidefiniteness of the mapping kel(1)implies Haussler’s theorem as a corollary,(2)exhibits an easy-to-check necessary and sufficient condition for mapping kels to be positive semideflnite,and(3)formalizes the mapping kel so that significant flexibility is provided in engineering new kels.As an evidence of the effectiveness of our results,we present a framework to engineer tree kels.The tree is a data structure widely used in many applications,and tree kels provide an effective method to analyze tree-type data.Thus,not only is the framework important as an example but also as a practical research tool.The description of the framework accompanies a survey of the tree kels in the literature,where we see that 18 out of the 19 surveyed tree kels of different types are instances of the mapping kel,and examples of novel interesting tree kels.