Edge Interface For Forest Iterators
Contents 
Introduction
We can define the concept of the forest iterator in terms of a general graph using the terminology from Boost graph  this is useful because at some point we may wish to generalize these constructs to apply to other graph structures.
A forest is (loosely) a Bidirectional Graph.
A forest iterator consists of the following:

v
, a vertex descriptor (in the forest implementation this is the pointer to the node). 
e
, anin_edge
index  this specifies how we are pointing at the node (this was the "leading bool" and is now theedge()
and the topic of this document). 
p
, a property (this is fixed to a single property of thevalue_type
of the iterator). 
f
, a function to move from anin_edge
to a correspondingout_edge
(for a fullorder iterator on a forest this is the identity function).
An edge is a directed link (with a from()
and to()
function)  and because it is bidirectional you can find the corresponding input link.
The increment operation can be defined as follows in pseudo code (once I'm more fluent in Boost Graph this could be described more concretely):
current_edge = out_edge(*v, e); v = to(curent_edge); e = f(find_in_edge(*v, current_edge));
In this way we can think of a forest iterator as being a traversal defined over a Bidirectional Graph. The edge()
interface on the iterator allows you to manipulate what in_edge
is being used to reach the vertex, and correspondingly what out_edge
will be followed next. The type of the result is the index for the edge sequence. In the case of forest this is an array, so std::size_t
is the correct type, even though it will only be a value of 0 or 1.
The signature for edge()
is:
std::size_t edge() const; // read access to the edge std::size_t& edge(); // write access to the edge
Child iterators
I think the same description holds for child_iterator
except f()
is replaced with trailing_of(*v)
and it no longer depends on which in_edge
you arrive on.
The reverse iterator problem with edge()
The standard notion of a reverse iterator transforms a range [first, last)
to the range (first, last]
without introducing any extra position by having the base iterator point one after the element. Reverse iterators for a strict sequence work fine with a forest, but a reverse iterator which supports edge()
is trickier.
The next (or prior) edge is dependent on where you came from and it is a function of a vertex. So to read the edge when your iterator is after the current mark you need to decrement
 but this means you are reading a vertex which is _before_ your offset range  this is because the edge is defined even on an end iterator (the vertex exists even if the property doesn't).
Right away then we violate our range just to read an edge()
 to set an edge you defer the actual assignment of the edge until you reach the node  then set the edge to determine where to go next  so incrementing a reverse iterator looks like:
 Decrement the base iterator
 Set the edge to our next edge
 remember what the next edge should be
We run into a big problem if we set the edge and then decrement the reverse iterator  setting the edge may have pivoted on the previous node and we should land on a node which isn't reachable in one step from the current base!
The algorithm for decrementing the reverse iterator is
 Decrement the base iterator
 Set the edge to our next edge
 increment the base iterator twice
 remember what the next edge should be
This all works on a forest, despite the range violations, because a forest is implemented as a loop structure and the prior vertex of any node is guaranteed to exist (that is, we have a root()
as well as an end()
and begin()
will yield root()
).
A simpler form of a reverse iterator could simply rely on the fact that the forest ranges are symmetrical. In that case rend()
could return root()
and we would scrap the whole off by one. This does not change the requirements of our reverse iterator at all, but for anyone familiar with STL reverse iterators it does change the behavior of base()
.
Final Resolution
The simpler form mentioned above was implemented and I've decided the simplifications are well worth it. The offbyone issue of base()
is resolved by simply having base()
return next(base_m)
. The requirement imposed by the reverse iterator is actually the same requirement imposed by set_next()
. For any valid range [first, last)
there must exist a valid range (prior(first), last)
.