If \(\alpha\) is the 2-core, \((\beta_0,\beta_1)\) the 2-quotient of a partition \(\lambda\), then the triple \((\alpha; \beta_0,\beta_1)\) uniquely determines \(\lambda\); see \textit{G. James} and \textit{A. Kerber} [The representation theory of the symmetric group (Addison-Wesley, Reading, MA) (1981; Zbl 0491.20010)]. The present author constructs a unique weighted binary tree representation for \(\lambda\) and provides various examples. Then he lists a series of problems concerning this correspondence. These ideas and their link with noncrossing partitions and Catalan numbers were prompted by work of Rodica Simion, to whose memory the present paper is dedicated.