This is a greedy forward selection algorithm for rotating the tree and looking for a better match.

This is useful for finding good trees for a tanglegram.

It goes through rotating dend1, then dend2, and so on - until a locally optimal solution is found.

Similar to "step1side", one tree is held fixed and the other tree is rotated. This function goes through all of the k number of clusters (from 2 onward), and each time rotates the branch which was introduced in the new k'th cluster. This rotated tree is compared with the fixed tree, and if it has a better entanglement, it will be used for the following iterations. Once finished the rotated tree is held fixed, and the fixed tree is now rotated. This continues until a local optimal solution is reached.

```
untangle_step_rotate_2side(
dend1,
dend2,
L = 1.5,
direction = c("forward", "backward"),
max_n_iterations = 10L,
print_times = dendextend_options("warn"),
k_seq = NULL,
...
)
```

- dend1
a dendrogram object. The one we will rotate to best fit dend2.

- dend2
a dendrogram object. The one we will rotate to best fit dend1.

- L
the distance norm to use for measuring the distance between the two trees. It can be any positive number, often one will want to use 0, 1, 1.5, 2 (see 'details' in entanglement).

- direction
a character scalar, either "forward" (default) or "backward". Impacts the direction of clustering that are tried. Either from 2 and up (in case of "forward"), or from nleaves to down (in case of "backward")

If k_seq is not NULL, then it overrides "direction".

- max_n_iterations
integer. The maximal number of times to switch between optimizing one tree with another.

- print_times
logical (TRUE), should we print how many times we switched between rotating the two trees?

- k_seq
a sequence of k clusters to go through for improving dend1. If NULL (default), then we use the "direction" parameter.

- ...
not used

A list with two dendrograms (dend1/dend2), after they are rotated to best fit one another.

```
if (FALSE) {
dend1 <- USArrests[1:20, ] %>%
dist() %>%
hclust() %>%
as.dendrogram()
dend2 <- USArrests[1:20, ] %>%
dist() %>%
hclust(method = "single") %>%
as.dendrogram()
set.seed(3525)
dend2 <- shuffle(dend2)
tanglegram(dend1, dend2, margin_inner = 6.5)
entanglement(dend1, dend2, L = 2) # 0.79
dend2_corrected <- untangle_step_rotate_1side(dend2, dend1)
tanglegram(dend1, dend2_corrected, margin_inner = 6.5) # Good.
entanglement(dend1, dend2_corrected, L = 2) # 0.0067
# it is better, but not perfect. Can we improve it?
dend12_corrected <- untangle_step_rotate_2side(dend1, dend2)
tanglegram(dend12_corrected[[1]], dend12_corrected[[2]], margin_inner = 6.5) # Better...
entanglement(dend12_corrected[[1]], dend12_corrected[[2]], L = 2) # 0.0045
# best combination:
dend12_corrected_1 <- untangle_random_search(dend1, dend2)
dend12_corrected_2 <- untangle_step_rotate_2side(dend12_corrected_1[[1]], dend12_corrected_1[[2]])
tanglegram(dend12_corrected_2[[1]], dend12_corrected_2[[2]], margin_inner = 6.5) # Better...
entanglement(dend12_corrected_2[[1]], dend12_corrected_2[[2]], L = 2) # 0 - PERFECT.
}
```