There is a lot of risk involved in tree surgery, and those who work in this industry need to understand the regulations and be compliant with all safe practice laws. This is very important for anyone who operates a tree surgery business or who is a tree surgeon working for themselves.
Tree surgeons are at risk from falling tree branches, falling trees, falls from a tree or other height, the dangerous equipment they use and the sawdust and other substances that they are exposed to.
The exhaustive list of tree work health and safety guidelines can be found on the Health & Safety Executive website.
According to the regulations of the Management of Health and Safety at Work Regulations 1999, all self-employed people and employers must ensure their workplace is assessed for risk. This extends to those places where tree surgery is performed. More information about risk assessment can be found on the HSE website. 
There are duties that have been imposed by the Work at Height Regulations 2005 that apply to anyone who works at a height. Any employers and self-employed individuals must ensure that their employees have been trained properly and are being supervised. They must also use climbing harnesses, ladders and other equipment that is designed for the work they are doing and that fits the person using it.
The Provision and Use of Work Equipment Regulations 1992 lays out the guidelines for using work equipment. There will be machinery that needs to be evaluated before it can be used or that those using it will need to be qualified for. For example, any tree surgeon that uses a chainsaw must show that they can do so competently. The HSE has provided a guide that details all the necessary information about this.
There may be separate requirements for various types of lifting equipment. All these will be covered in the Lifting Operations and Lifting Equipment Regulations 1998. You can learn more about the lifting requirements by reading the information on the HSE website.
Under the Personal Protective Equipment at Work Regulations 1992, certain kinds of jobs will require personal protective equipment. All tree surgeons will need to wear the right equipment, including helmets, gloves, boots and eye protection as they work. There are guidelines for specialised equipment to be worn when operating certain machinery, such as the chainsaw.
All this is covered on the HSE website.
When dealing with potentially hazardous substances, arborists must protect both themselves and their customers. Tree surgeons may deal with sawdust and petrol fumes regularly, and the regulations concerning those substances is covered in the Control of Substances Hazardous to Health Regulations 2002 (COSHH).
The complexity of the health and safety regulations means that they need to be studied and researched extensively. HSE has some resources that can help explain what the requirements are for the different aspects of tree surgery.