[Darren Abbey]

Backcrossing works when you're trying to transfer a single or few dominant traits from one line into another, with relatively few other impacts. This works for dominant traits because you can see them in the progeny at every stage. Recessive traits from the donor line will be hidden in the F1s and will remain hidden in the progeny of any backcrosses to the recipient line.

If you don't know the genetics of the traits you're interested in, you will have to grow out a reasonable number of progeny from the selfed F1s in order to characterize the genetics in the F2s. Growing 20 or so F2s from a selfed-F1 might not give you every combination you might expect, but it will allow you to examine the ratios of each of the traits you're interested in.
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The color genetics are relatively known, so we can examine them in detail:

Color genetics involved in your cross:
P : (wwgg) green "Tatume" x (wwGG) yellow "Lemon"
F1 : (wwGg) yellow
F2 : 3 (1 wwGG; 2 wwGg) yellow; 1 (wwgg) green

F1 x "Tatume" backcross:
P : (wwGg) yellow x (wwgg) green "Tatume"
F1 : (wwGg) yellow + (wwgg) green

Recurrent backcrosses of yellow progeny to "Tatume" will never clear out the recessive allele for green, but will convert more and more of other loci in the crossed line to the versions found in Tatume. After several generations (maybe 8? the more the better.) you will have produced a squash very much like "Tatume", except if will have the single selected dominant allele. At this time, selfing the line can then be used to produce a double-Dominant version.

"Tatume2" x "Tatume2":
P : (wwGg) yellow "Tatume2" x (wwGg) yellow "Tatume2"
F1 : 3 (1 wwGG; 2 wwGg) yellow; 1 (wwgg) green

At this stage, you will have to self each of the F1s and screen each set of F2s separately. 2:3 of the yellow F1s will produce green F2s… but that 1:3 won't. That 1:3 represents your final strain, "Yellow Tatume".

If you select for multiple traits at each step, you will generate a version of "Tatume" differing from the original only in those traits. Recessive traits in your final strain will match the version in "Tatume". The genetics of flavor and production are likely complicated and may involve recessive alleles. You will have to examine a batch of those first selfed-F1 F2s to determine this.
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If you're trying to carry over recessive traits using backcrossing, you will have to examine F2s at every step to ensure the backcrossed progeny you're working with still carry the recessive versions of the genes.

Hypothetical recessive flavor genetics:
P : (AA) bitter "Tatume" x (aa) sweet "Lemon"
F1 : (Aa) bitter
F2 : 3 (1 AA; 2 Aa) bitter; 1 (aa) sweet

F1 x "Tatume" backcross:
P : (Aa) bitter x (AA) bitter "Tatume"
F1 : (AA) bitter + (Aa) bitter

At this stage, you won't know which to use. You will have to grow out F2s from each to identify which line still has the 'sweet' allele. The more cycles of this, the more the resulting progeny will be like the original "Tatume", with the exception of the one recessive trait that you kept screening for.

"Tatume3" x "Tatume3":
P : (Aa) bitter "Tatume3" x(Aa) bitter "Tatume3"
F1 : 3 (1 AA; 2 Aa) bitter; 1 (aa) sweet

The final step of getting rid of that bitter allele from the original "Tatume" can then be done in one generation, giving you "Sweet Tatume".
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Backcrossing to carry over a single recessive trait will take roughly twice as many years as for a dominant trait.

If you screen large numbers of F2s from the original "Tatume" x "Lemon" cross, you might find what you're looking for in the second year after starting. If you don't find it that second year, you will have a much better idea about the genetics of the traits you're interested in and can then plan how many more F2s you will need to grow to have good odds of finding it.

I prefer the idea of screening lots of F2s because you can find combinations of traits that were not obvious in the parents that result in a progeny plant which you like even better than either original plant.