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In this paper, we investigate the weighted Catalan, Motzkin and Schr\"odernumbers together with the corresponding weighted paths. The relation betweenthese numbers is illustrated by three equations, which also lead to some knownand new interesting identities. To show these three equations, we providecombinatorial proofs. One byproduct is to find a bijection between two sets ofCatalan paths: one consisting of those with $k$ valleys, and the otherconsisting of $k$ $\mathbf{N}$ steps in even positions.

Identities involving weighted Catalan, Schroder and Motzkin paths