Hierarchical clustering: consistent and dichotomous group names representing hierarchy within tree - r

Question

I aim at producing a typology of sites based through hierarchical clustering of species abundance data. Therefore, I successively cut the dendrogram into 2, 3, 4 ... z groups.
The cluster group names are automatically attributed by the function cutree() representing numbers from 1 to z in a non-consistent manner. For instance, in a clustering with three groups, "group 2" may not correspond to "group 2" in a clustering with six groups. This makes interpreting the dendrogram very difficult.
The code below provides a reproducible example. It produces a hierarchical clustering of 50 observations and successively cuts the dendrogram in a for loop. The final output data frame 'cluster.grps' contains the cluster group affiliation for each obervation and successive cutting height (HC_2 = hierarchical clustering with 2 groups; HC_3 = hc with three groups; etc.).
set.seed(1)
data <- data.frame(replicate(10,sample(0:10,50,rep=TRUE))) # create random site x species dataframe
clust <- hclust(dist(data), method = "ward.D") # implement hierarchical clustering
# Set maximum number of groups
z <- 6
# Loop for successive tree cutting
lst <- list()
for (i in 2:z) {
# Slicing the dendrogram
cutree <- cutree(clust, k = i) # k = number of groups
lst[[(i-1)]] <- cutree
}
names(lst) <- 2:(z-1)
cluster.grps <- as.data.frame(lst)
colnames(cluster.grps) <- paste("HC",as.character(2:(z)),sep ="_")
I now wish to attribute dichotomous names that represent the level of hierarchy in the tree: 1, 2 for the first level; 1.1, 1.2, 2.1, 2.2 for the second level; 1.1.1, 1.1.2, 1.2.1, 1.2.2, etc. for the third level and so on.
Ideally, the table 'cluster.grps' would look like this:
Site
HC_2
HC_3
HC_4
Site 1
1
1.1
1.1
Site 2
2
2
2.1
Site 3
1
1.2
1.2
Site 4
2
2
2.2
My first thought was to code nested clusterings in which I start with a first clustering of all observations into two groups and subsequently splitting each group of the first clustering independently into two consecutive groups, yielding four groups at the second hierarchical level. This requires quite a long code, though and I was wondering whether there might be a more elegant way.
Any thoughts?