Right, I got curious about this so I did some Google-fu.
The basic components of a traditional Crystal Tree/Garden are:
- Laundry Bluing: a suspension of blue dye in water traditionally added to laundry to make textiles appear whiter(!)
- Table salt (probably Sodium Chloride)
- Household ammonia (with no soap)
- Non-porous Bowl
- Porous growing material
Crystal growing was popularised in the Great Depression. People traditionally used Mrs Stuart's Bluing. Obviously, this didn't escape the company's attention and to that end, they sell crystal growing kits and have an instruction page on their website. Taken from that page:
Mrs. Stewart’s Bluing is a colloidal suspension of extremely minute particles of blue powder (Ferric Hexacyanoferrate). This is not a solution in the true chemical meaning of that word.
As the water from the bluing and the clear water which is first added evaporate, two things happen. The blue particles can no longer be supported and the excess salt cannot stay in solution. The salt crystallization process will take place around the blue particles as nuclei, in much the same way as silver iodide cloud seeding accelerates the formation of rain drops.
... The purpose of the porous material (sponge pieces) is to provide a means for capillary action to carry the liquid containing bluing and salt up from the main source of liquid. This further speeds up evaporation and causes the crystals to form over a larger area than just the rim of the bowl.
What interested me about this question was the illusion of a tree, continually growing. Whilst our poster wasn't looking the family must have added (again from the website) more bluing, salt and water.
Causing the 'tree' to
“bloom” indefinitely into beautiful rosebuds, coral and crystal.
Now I would suspect that the green like growth was due to the family member using a 'greening', i.e., a suspension of green dye in water, rather than blue.
ADDED: Here's an example of a commercially available crystal tree growing kit. It's not as elegant as the poster's, but it serves the point.