People don't usually have a choice about becoming homeless, but their environment will affect how they respond to their homelessness. It's a cliché that some homeless people in America will choose to travel to California or Florida for the warmer climate (whether that's true or not I have no idea - would be interested in seeing data on that subject).

If someone becomes homeless in a place where homelessness has a significantly increased risk of death (such as freezing to death in Berlin), then either they move away, they engage with local support services, or they carry on as they are, risking death (and some of them will die as a result). Either way, the effect is a reduction in the number of homeless people at that location.

Thankfully a lot of places recognise some of the risks and try to address them. In the UK, many cities have a cold-weather response: when the temperature drops below Xº, additional shelters are opened, and street-teams look for people sleeping rough and guide them towards the shelters.

Well, at least in Germany there's virtually no homelessness due to a lack of resources. The government has to provide sufficient housing. The reasons for homelessness around here are most of the time different ones than "can't afford to pay rent". In many cases mental health issues and bureaucratic hurdles which might be too high to tackle for those stuck in a difficult situation. It's mostly down to having Street Workers which are able to connect to those people and help them doing the first steps into the right direction.