I do not want to write .d.ts files for every single external javascript file I want to use, is that seriously what Typescript wants me to do?

No. The simplest / quickest solution is simply to tell it that there is some variable Tree out there. This is as simple as:

declare var Tree:any; // Magic
var myTree = Tree.tree({})

TypeSafety is a sliding scale in TypeScript. In this case you are only telling the compiler that there is something called Tree that you will manage and don't care for much typesafety beyond the fact that it is there.

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IMHO: The line declare var Tree:any; is much simpler syntax than other JS veficiation tools would have you write to declare your usage of variables that are not present in your code.

Update

interface ITree {
    .. further methods and properties...
}

interface ITreeFactory {
    tree(input: { [key: string]: any }): Itree
};

declare var Tree: ITreeFactory; // magic...