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If you use ggdendrogram() to create your plot, the resulting object is a ggplot. You have full control over this using any function available in ggplot.

First create an example dataset.

library(ggdendro)
library(ggplot2)
hc <- hclust(dist(USArrests), "ave")

Plot the default ggdendrogram() output:

ggdendrogram(hc, rotate = FALSE, size = 2)

Use a different theme:

ggdendrogram(hc, rotate = FALSE, size = 2) +
  theme_bw()

Or modify just one element, for example add a y-axis.

ggdendrogram(hc, rotate = FALSE, size = 2) +
  theme( axis.line.y = element_line() )

In summary, ggdendrogram() is a convenience function that creates a ggplot. Once you have this plot, you can modify the plot using tools that you are familiar with.