A balanced tree-like presentation of $S_3$

Campbell, C. M.; Mitchell, J. D.; Ruškuc, N. On defining groups efficiently without using inverses. Math. Proc. Cambridge Philos. Soc. 133 (2002), no. 1, 31–36 shows among other things that a group with a balanced group presentation has a balanced semigroup presentation. The proof gives a tree like presentation. Look at the proof of Prop 2.4 and 2.5 and note R' and R'' are empty when the group presentation was balanced. See https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/on-defining-groups-efficiently-without-using-inverses/ED8A580F336B47B2162DB5E3A5FA8459

I see that Jeremy has beaten me to it - but here is a solution found by computer by choosing random entries from the group multiplication table.

This one also generates $S_3$ as a semigroup. The Magma command $\mathtt{RWSMonoid}$ applies the Knuth-Bendix algorithm to the presentation, and regards it as a monoid presentation. It has order $7$ as a monoid with identity equal to the empty word, and so it has order $6$ as a semigroup presentation.

$$\langle a,b,c,d,e,f\mid a=f^2,b=ab,c=df,d=ec,e=db,f=ce\rangle.$$

  > G := Group< a,b,c,d,e,f | a = f^2, b = a*b, c = d*f, d=e*c, e=d*b, f=c*e >;
  > Order(G);
  6
  > Homomorphisms(G,Sym(3))[1];
   Homomorphism of GrpFP: G into GrpPerm: X, Degree 3, Order 2 * 3 induced by
      G.1 |--> Id(X)
      G.2 |--> (1, 2, 3)
      G.3 |--> (1, 3, 2)
      G.4 |--> (1, 2)
      G.5 |--> (1, 3)
      G.6 |--> (2, 3)

> M := Monoid< a,b,c,d,e,f | a = f^2, b = a*b, c = d*f, d=e*c, e=d*b, f=c*e >; 
> R := RWSMonoid(M);
> Order(R);
7