Who is responsible for maintaining the garden in a tenancy?

There are no hard and fast rules regarding garden maintenance within a tenancy, but generally speaking there is often a division of responsibilities laid out in the tenancy agreement. 
A tenant will commonly be expected to keep up with some of the more low-skill gardening maintenance tasks such as weeding and lawn mowing. However, the standard of presentation in the garden will often slip if it is left to the tenants, so some landlords will offer to cover this themselves as a ‘perk’. Sometimes the landlord meets the cost of this, while others may factor the cost into the value of the rental payments. 
Generally speaking, it is not a good idea to assign the maintenance of more involved gardening work to the tenants, such as keeping trees, hedges and shrubs in check, and landlords should take responsibility for this aspect of the property management. Typically, most tenants do not have the skills to maintain these parts of the garden effectively, and may end up doing more harm than good if they attempt to carry out work they are not experienced in. There is also the risk of a tenant unwittingly damaging or removing all or parts of protected species trees, which is against the law. 
Landlords have full responsibility for ensuring that the trees, hedges and shrubs on their property remain in a safe condition, and must take action if any are posing a threat to safety, for example as a result of storm damage, or growing too near to power lines.
A low maintenance garden is a big draw card for good tenants, who are often busy people and may not have the time or motivation to maintain the garden to a high enough standard. Most landlords are busy people too, so many find the most effective solution is to hire the regular services of contract gardeners and/or arborists to maintain the garden’s trees and hedgerows.
It is a good idea for landlords to take photographs of the garden when it is a condition they wish to maintain, and supply these to the contractors as a visual record. A qualified arborist will ensure that all dead wood is continually removed and that trees are regularly pruned to be kept away from windows, drains etc. Depending on the location of the trees, regular pruning may be necessary to allow adequate light into the house and garden. This is in the best interests of both the tenants and the landlords, as it will allow the shrubbery and lawns around it to grow properly, and will reduce the chance of problems like damp and mould from forming inside the property.
Renters are more likely look after their garden space in a long term tenancy, as long term tenants will usually see the property as their home rather than a mere base, and be more inclined to take pride in its appearance. If landlords wish to stipulate in the tenancy agreement that some or all of the garden maintenance should be the responsibility of those renting, then it is wise to consider offering a 12 month contract or longer to have the best chance of finding a compliant tenant.
Finally, landlords should take into account the effect that the appearance of the garden and the health of the trees and hedgerows within it will have upon the value of their property, as well as their relations with the local neighbours and possibly the council, when weighing up whether they should assign the responsibility for tree and hedgerow maintenance to the tenants or a qualified arborist.