Consider the following question: Let $X$ be a compact complex manifold and $\beta \in H_2(X, \mathbb{Z})$ a fixed homology class. Let $\mu_1, \mu_2, \ldots, \mu_k$ denote certain generic submanifolds (of the right dimensions) in $X$. What is $N_{g,\beta}(\mu_1, \ldots, \mu_k)$, the number of genus $g$ curves in $X$ representing the class $\beta$ and intersecting the submanifolds $\mu_i$?

If $X:= \mathbb{P}^2$, then there is a complete formula for $N_{g,d}(p_1, \ldots, p_{3d-1+g})$, the number of genus $g$ degree $d$ curves through $3d-1+g$ generic points, using Tropical Geometry (and also using the Caporaso Harris formula). In fact I think if $X$ is any toric two dimensional variety there is a formula for $N_{g, \beta}$ using Tropical Geometry.

$\textbf{Question:}$ Is there a $\textit{Tropical geometric}$ solution
for computing $N_{g,d}$ for $\mathbb{P}^n$ where $n>2$? Kontsevich's derivation
of $N_{0,d}$,
the number of degree $d$ rational curves in $\mathbb{P}^2$ generalizes for any

$\mathbb{P}^n$. More generally, is there a tropical geometric solution
(or actually any solution) for $N_{g, \beta}$ on complex manifolds $X$ other
than $\mathbb{P}^n$ (my question here primarily being for $g>0$, although I
am interested in knowing what is known for $g=0$ as well.)

$\textbf{Remark:}$ One of the reasons for asking this question is as follows (the reason/hope might be very naive): I am just wondering if using any of these enumerative results one can make some partial (but direct) verification of some predictions made by Mirror Symmetry (particularly higher genus mirror symmetry predictions). To take one example, it is predicted in this paper (page 34)

http://arxiv.org/pdf/hep-th/0612125v1.pdf

that the number of degree $4$, genus $2$ curves on the quintic three fold is $534750$. Is it conceivable that there could be a direct way to see that?

$\textbf{Added Later:}$ My last comment (about degree $4$ genus $2$ curves on the quintic threefold) is incorrect as has been pointed out. I was very naively concluding that the numbers on page $34$ (at least the initial ones) were enumerative, which they are not (the authors don't claim that either). My hope that using enumerative formulas for genus g curves one can explain some of these BPS numbers is perhaps very naive. I am nevertheless interested to know if there are enumerative formulas for genus g curves into manifolds with dimension greater than two (irrespective of whether they have any use in explaining the numbers obtained from Mirror Symmetry).