  Leibniz algebras can be seen as a ``non-commutative" analogue of Lie
algebras. Nijenhuis operators on Leibniz algebras introduced by Cari\~{n}ena,
Grabowski, and Marmo in [J. Phys. A: Math. Gen. 37(2004)] are (1, 1)-tensors
with vanishing Nijenhuis torsion. Recently triangular Leibniz bialgebras were
introduced by Tang and Sheng in [J. Noncommut. Geom. 16(2022)] via the twisting
theory of twilled Leibniz algebras. In this paper we find that Leibniz algebras
are very closely related to Nijenhuis operators, and prove that a triangular
symplectic Leibniz bialgebra together with a dual triangular structure must
possess Nijenhuis operators, which makes it possible to study the applications
of Nijehhuis operators from the perspective of Leibniz algebras. At the same
time, we regain the classical Leibniz Yang-Baxter equation by using the tensor
form of classical $r$-matrics. At last we give the classification of triangular
Leibniz bialgebras of low dimensions.
