This is a good question. Firstly, let's get the terminology straightened out: The terms recessive and dominant are not actually used to describe genes, but rather alleles. The term allele is used for alternative forms of the same gene.(1) Many genes have two or more alleles, and some genes have alleles that can be described as dominant or recessive.
It is also useful to be familiar with the terms genotype and phenotype. Genotype refers to the form of the genetic material (DNA), while phenotype refers to the observed effect of the genotype.
Secondly, it's important to realize that the concept of "dominant" and "recessive" is a human-made abstraction. The concept of dominant and recessive genes were invented (or discovered, if you like) before we reached a good undertanding of how genetics works at the molecular level. Thus, genetics in the early days had a much cruder understanding of how different phenotypes depend on the genotype.
The dominant/recessive concept is useful in cases where there is a limited number of possible allele pairs and one or several alleles can mask the effect of other alleles. This is a very limited set of cases. Many genes have many different alleles, and it's hard to account for all of them. Many alleles also do not have a clear observable effect, in which case any "masking" effect is not prominent. To quote Wikipedia (2):
The most common basis of dominance and recessiveness is that the
dominant allele codes for a functional protein and the recessive
allele for a mutant, non-functional protein.
A further important point is made in the following passage:
Dominance is not inherent to an allele. It is a relationship between
alleles; one allele can be dominant over a second allele, recessive to
a third allele, and codominant to a fourth. Dominance should be
distinguished from epistasis, a relationship in which an allele of one
gene affects the expression of an allele at a different gene.
Let's consider your example: A single gene with two different alleles, T and t. Let's assume the T allele codes for a more powerful version of the growth hormone than the t allele, or that the t allele produces a non-functional hormone. Furthermore, we assume that production of the T hormone from a single allele is enough to make the plant grow tall, and that production of T or t from a second allele has no further effect. Then, we can say that the T allele masks the effect of the t allele. However, as you can probably imagine, one allele fully masking the effect of another is rather rare. So recessive/dominant allele pairs are rather the exception than the norm.
Thus, the that place for the two alleles in the genome of the plant is definitely not empty. Note that a mutation could in principle cause one of the genes to be deleted in its entirety from one of the chromosomes of the plant during reproduction. The "place" for the gene on that chromosome would then indeed be empty. This has nothing to do with how dominance and recessiveness usually works, except that a recessive gene could become expressed if the dominant allele is deleted.
So, at the molecular level there is no concept of recessive or dominant alleles. Some genes have allele pairs that can be described as dominant/recessive in relation to each other, most others do not. The effect of different combinations of alleles determines whether the relationship between the alleles is described in terms of dominant/recessive.